ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
OF
PROPER
HYPERBOLIC
CURVES
Shinichi
Mochizuki
June
2007
In
this
paper,
we
develop
the
theory
of
“cuspidalizations”
of
the
étale
fundamental
group
of
a
proper
hyperbolic
curve
over
a
finite
or
nonarchimedean
mixed-characteristic
local
field.
The
ultimate
goal
of
this
theory
is
the
group-theoretic
reconstruction
of
the
étale
fundamental
group
of
an
arbitrary
open
subscheme
of
the
curve
from
the
étale
fundamental
group
of
the
full
proper
curve.
We
then
apply
this
theory
to
show
that
a
certain
absolute
p-adic
version
of
the
Grothendieck
Conjec-
ture
holds
for
hyperbolic
curves
“of
Belyi
type”.
This
includes,
in
particular,
affine
hyperbolic
curves
over
a
nonarchimedean
mixed-characteristic
local
field
which
are
defined
over
a
number
field
and
isogenous
to
a
hyperbolic
curve
of
genus
zero.
Also,
we
apply
this
theory
to
prove
the
analogue
for
proper
hyperbolic
curves
over
finite
fields
of
the
version
of
the
Grothendieck
Conjecture
that
was
shown
in
[Tama].
Contents:
§0.
Notations
and
Conventions
§1.
Maximal
Abelian
Cuspidalizations
§2.
Points
and
Functions
§3.
Maximal
Pro-l
Cuspidalizations
Appendix:
Free
Lie
Algebras
Introduction
Let
X
be
a
proper
hyperbolic
curve
over
a
field
k
which
is
either
finite
or
nonarchimedean
local
of
mixed
characteristic;
let
U
⊆
X
be
an
open
subscheme
of
X.
Write
Π
X
for
the
étale
fundamental
group
of
X.
In
this
paper,
we
study
the
extent
to
which
the
étale
fundamental
group
of
U
may
be
group-theoretically
reconstructed
from
Π
X
.
In
§1,
we
show
that
the
abelian
portion
of
the
extension
of
Π
X
determined
by
the
étale
fundamental
group
of
U
may
be
group-theoretically
reconstructed
from
Π
X
[cf.
Theorem
1.16,
(iii)],
and,
moreover,
that
this
construction
has
certain
remarkable
rigidity
properties
[cf.
Propositions
1.15,
(i);
2.6,
(i)].
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
In
§2,
we
show
that
this
abelian
portion
of
the
extension
is
sufficient
to
recon-
struct
[in
essence]
the
multiplicative
group
of
the
function
field
of
X
[cf.
Theorem
2.5,
(ii)].
In
the
case
of
nonarchimedean
[mixed-characteristic]
local
fields,
this
already
implies
various
interesting
consequences
in
the
context
of
the
absolute
an-
abelian
geometry
studied
in
[Mzk5],
[Mzk6],
[Mzk8].
In
particular,
it
implies
that
the
absolute
p-adic
version
of
the
Grothendieck
Conjecture
[i.e.,
an
absolute
version
of
[a
certain
portion
of]
the
relative
result
that
appears
as
the
main
result
of
[Mzk4]]
holds
for
hyperbolic
curves
“of
Belyi
type”
[cf.
Definition
2.9;
Corollary
2.12].
This
includes,
in
particular,
hyperbolic
curves
“of
strictly
Belyi
type”,
i.e.,
affine
hy-
perbolic
curves
over
a
nonarchimedean
[mixed-characteristic]
local
field
which
are
defined
over
a
number
field
and
isogenous
to
a
hyperbolic
curve
of
genus
zero.
In
particular,
we
obtain
a
new
countable
class
of
“absolute
curves”
[in
the
terminology
of
[Mzk6]],
whose
absoluteness
is,
in
certain
respects,
reminiscent
of
the
absolute-
ness
of
the
canonical
curves
of
p-adic
Teichmüller
theory
discussed
in
[Mzk6]
[cf.
Remark
2.13.1],
but
[in
contrast
to
the
class
of
canonical
curves]
appears
[at
least
from
the
point
of
view
of
certain
circumstantial
evidence]
unlikely
to
be
Zariski
dense
in
most
moduli
spaces
[cf.
Remark
2.13.2].
Finally,
in
§3,
we
apply
the
theory
of
the
weight
filtration
[cf.,
e.g.,
[Kane],
[Mtm]],
together
with
various
generalities
concerning
free
Lie
algebras
[cf.
the
Appendix],
to
develop
various
“higher
order
generalizations”
of
the
theory
of
§1,
2.
In
particular,
we
obtain
various
“higher
order
generalizations”
of
the
“remarkable
rigidity”
referred
to
above
[cf.
Propositions
3.7,
3.9,
especially
Proposition
3.9,
(iii)],
which
we
apply
to
show
that,
relative
to
the
notation
introduced
above,
the
geometrically
pro-l
portion
[where
l
is
a
prime
number
invertible
in
k]
of
the
étale
fundamental
group
of
U
may
be
recovered
from
Π
X
,
at
least
when
U
is
obtained
from
X
by
removing
a
single
k-rational
point
[cf.
Theorem
3.10].
This,
along
with
the
theory
of
§2,
allows
one
to
verify
the
analogue
for
proper
hyperbolic
curves
over
finite
fields
of
the
version
of
the
Grothendieck
Conjecture
that
was
shown
in
[Tama]
[cf.
Theorem
3.12].
Acknowledgements:
I
would
like
to
thank
Akio
Tamagawa,
Makoto
Matsumoto,
and
Seidai
Yasuda
for
various
useful
comments.
Also,
I
would
like
to
thank
Yuichiro
Hoshi
for
his
careful
reading
of
an
earlier
version
of
this
manuscript,
which
led
to
the
discovery
of
various
errors
in
this
earlier
version.
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
3
Section
0:
Notations
and
Conventions
Numbers:
the
profinite
completion
of
the
additive
group
of
rational
We
shall
denote
by
Z
integers
Z.
If
p
is
a
prime
number,
then
Z
p
denotes
the
ring
of
p-adic
integers;
Q
p
denotes
its
quotient
field.
We
shall
refer
to
as
a
p-adic
local
field
(respectively,
nonarchimedean
local
field)
any
finite
field
extension
of
Q
p
(respectively,
a
p-adic
local
field,
for
some
p).
A
number
field
is
defined
to
be
a
finite
extension
of
the
field
of
rational
numbers.
If
Σ
is
a
set
of
prime
numbers,
then
we
shall
refer
to
a
positive
integer
each
of
whose
prime
factors
belongs
to
Σ
as
a
Σ-integer.
We
shall
refer
to
a
finite
étale
covering
of
schemes
whose
degree
is
a
Σ-integer
as
a
Σ-covering.
Also,
we
shall
write
Primes
for
the
set
of
all
prime
numbers.
Topological
Groups:
Let
G
be
a
Hausdorff
topological
group,
and
H
⊆
G
a
closed
subgroup.
Let
us
write
G
ab
for
the
abelianization
of
G
[i.e.,
the
quotient
of
G
by
the
closed
subgroup
of
G
topologically
generated
by
the
commutators
of
G].
Let
us
write
def
Z
G
(H)
=
{g
∈
G
|
g
·
h
=
h
·
g,
∀
h
∈
H}
for
the
centralizer
of
H
in
G;
N
G
(H)
=
{g
∈
G
|
g
·
H
·
g
−1
=
H}
def
for
the
normalizer
of
H
in
G;
and
def
C
G
(H)
=
{g
∈
G
|
(g
·
H
·
g
−1
)
H
has
finite
index
in
H,
g
·
H
·
g
−1
}
for
the
commensurator
of
H
in
G.
Note
that:
(i)
Z
G
(H),
N
G
(H)
and
C
G
(H)
are
subgroups
of
G;
(ii)
we
have
inclusions
H,
Z
G
(H)
⊆
N
G
(H)
⊆
C
G
(H)
and
(iii)
H
is
normal
in
N
G
(H).
If
H
=
N
G
(H)
(respectively,
H
=
C
G
(H)),
then
we
shall
say
that
H
is
normally
terminal
(respectively,
commensurably
terminal)
in
G.
Note
that
Z
G
(H),
N
G
(H)
are
always
closed
in
G,
while
C
G
(H)
is
not
necessarily
closed
in
G.
If
G
1
,
G
2
are
Hausdorff
topological
groups,
then
an
outer
homomorphism
G
1
→
G
2
is
defined
to
be
an
equivalence
class
of
continuous
homomorphisms
G
1
→
G
2
,
where
two
such
homomorphisms
are
considered
equivalent
if
they
differ
4
SHINICHI
MOCHIZUKI
by
composition
with
an
inner
automorphism
of
G
2
.
The
group
of
outer
automor-
phisms
of
G
[i.e.,
bijective
bicontinuous
outer
homomorphisms
G
→
G]
will
be
denoted
Out(G).
If
G
is
center-free,
then
there
is
a
natural
exact
sequence:
1
→
G
→
Aut(G)
→
Out(G)
→
1
[where
the
homomorphism
G
→
Aut(G)
is
defined
by
letting
G
act
on
G
by
conju-
gation].
If
G
is
a
profinite
group
such
that,
for
every
open
subgroup
H
⊆
G,
we
have
Z
G
(H)
=
{1},
then
we
shall
say
that
G
is
slim.
One
verifies
immediately
that
G
is
slim
if
and
only
if
every
open
subgroup
of
G
is
center-free
[cf.
[Mzk5],
Remark
0.1.3].
If
G
is
a
profinite
group
and
Σ
is
set
of
prime
numbers,
then
we
shall
say
that
G
is
a
pro-Σ
group
if
the
order
of
every
finite
quotient
group
of
G
is
a
Σ-integer.
If
Σ
=
{l}
is
of
cardinality
one,
then
we
shall
refer
to
a
pro-Σ
group
as
a
pro-l
group.
Curves:
Suppose
that
g
≥
0
is
an
integer.
Then
if
S
is
a
scheme,
a
family
of
curves
of
genus
g
X
→
S
is
defined
to
be
a
smooth,
proper,
geometrically
connected
morphism
of
schemes
X
→
S
whose
geometric
fibers
are
curves
of
genus
g.
Suppose
that
g,
r
≥
0
are
integers
such
that
2g
−
2
+
r
>
0.
We
shall
denote
the
moduli
stack
of
r-pointed
stable
curves
of
genus
g
(where
we
assume
the
points
to
be
unordered)
by
M
g,r
[cf.
[DM],
[Knud]
for
an
exposition
of
the
theory
of
such
curves;
strictly
speaking,
[Knud]
treats
the
finite
étale
covering
of
M
g,r
determined
by
ordering
the
marked
points].
The
open
substack
M
g,r
⊆
M
g,r
of
smooth
curves
will
be
referred
to
as
the
moduli
stack
of
smooth
r-pointed
stable
curves
of
genus
g
or,
alternatively,
as
the
moduli
stack
of
hyperbolic
curves
of
type
(g,
r).
A
family
of
hyperbolic
curves
of
type
(g,
r)
X
→
S
is
defined
to
be
a
morphism
which
factors
X
→
Y
→
S
as
the
composite
of
an
open
immersion
X
→
Y
onto
the
complement
Y
\D
of
a
relative
divisor
D
⊆
Y
which
is
finite
étale
over
S
of
relative
degree
r,
and
a
family
Y
→
S
of
curves
of
genus
g.
One
checks
easily
that,
if
S
is
normal,
then
the
pair
(Y,
D)
is
unique
up
to
canonical
isomorphism.
(Indeed,
when
S
is
the
spectrum
of
a
field,
this
fact
is
well-known
from
the
elementary
theory
of
algebraic
curves.
Next,
we
consider
an
arbitrary
connected
normal
S
on
which
a
prime
l
is
invertible
(which,
by
Zariski
localization,
we
may
assume
without
loss
of
generality).
Denote
by
S
→
S
the
finite
étale
covering
parametrizing
orderings
of
the
marked
points
and
trivializations
of
the
l-torsion
points
of
the
Jacobian
of
Y
.
Note
that
S
→
S
is
independent
of
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
5
the
choice
of
(Y,
D),
since
(by
the
normality
of
S),
S
may
be
constructed
as
the
normalization
of
S
in
the
function
field
of
S
(which
is
independent
of
the
choice
of
(Y,
D)
since
the
restriction
of
(Y,
D)
to
the
generic
point
of
S
has
already
been
shown
to
be
unique).
Thus,
the
uniqueness
of
(Y,
D)
follows
by
considering
the
classifying
morphism
(associated
to
(Y,
D))
from
S
to
the
finite
étale
covering
of
(M
g,r
)
Z[
1
l
]
parametrizing
orderings
of
the
marked
points
and
trivializations
of
the
l-torsion
points
of
the
Jacobian
[since
this
covering
is
well-known
to
be
a
scheme,
for
l
sufficiently
large].)
We
shall
refer
to
Y
(respectively,
D;
D)
as
the
compactification
(respectively,
divisor
of
cusps;
divisor
of
marked
points)
of
X.
A
family
of
hyperbolic
curves
X
→
S
is
defined
to
be
a
morphism
X
→
S
such
that
the
restriction
of
this
morphism
to
each
connected
component
of
S
is
a
family
of
hyperbolic
curves
of
type
(g,
r)
for
some
integers
(g,
r)
as
above.
A
family
of
hyperbolic
curves
X
→
S
of
type
(0,
3)
will
be
referred
to
as
a
tripod.
If
X
is
a
hyperbolic
curve
over
a
field
K
with
compactification
X
⊆
X,
then
we
shall
write
X
cl
;
X
cl+
for
the
sets
of
closed
points
of
X
and
X,
respectively.
If
X
K
(respectively,
Y
L
)
is
a
hyperbolic
curve
over
a
field
K
(respectively,
L),
then
we
shall
say
that
X
K
is
isogenous
to
Y
L
if
there
exists
a
hyperbolic
curve
Z
M
over
a
field
M
together
with
finite
étale
morphisms
Z
M
→
X
K
,
Z
M
→
Y
L
.
Note
that
in
this
situation,
the
morphisms
Z
M
→
X
K
,
Z
M
→
Y
L
induce
finite
separable
inclusions
of
fields
K
→
M
,
L
→
M
.
[Indeed,
this
follows
immediately
from
the
×
easily
verified
fact
that
every
subgroup
G
⊆
Γ(Z,
O
Z
)
such
that
G
{0}
determines
a
field
is
necessarily
contained
in
M
×
.]
If
X
is
a
generically
scheme-like
algebraic
stack
[i.e.,
an
algebraic
stack
which
admits
a
“scheme-theoretically”
dense
open
that
is
isomorphic
to
a
scheme]
over
a
field
K
of
characteristic
zero
that
admits
a
[surjective]
finite
étale
[or,
equivalently,
finite
étale
Galois]
covering
Y
→
X,
where
Y
is
a
hyperbolic
curve
over
a
finite
extension
of
K,
then
we
shall
refer
to
X
as
a
hyperbolic
orbicurve
over
K.
[Although
this
definition
differs
from
the
definition
of
a
“hyperbolic
orbicurve”
given
in
[Mzk6],
Definition
2.2,
(ii),
it
follows
immediately
from
a
theorem
of
Bundgaard-Nielsen-Fox
[cf.,
e.g.,
[Namba],
Theorem
1.2.15,
p.
29]
that
these
two
definitions
are
equivalent.]
If
X
→
Y
is
a
dominant
morphism
of
hyperbolic
orbicurves,
then
we
shall
refer
to
X
→
Y
as
a
partial
coarsification
morphism
if
the
morphism
induced
by
X
→
Y
on
associated
coarse
spaces
[cf.,
e.g.,
[FC],
Chapter
I,
§4.10]
is
an
isomorphism.
Let
X
be
a
hyperbolic
orbicurve
over
an
algebraically
closed
field
of
character-
istic
zero;
denote
its
étale
fundamental
group
by
Δ
X
.
We
shall
refer
to
the
order
of
the
[manifestly
finite!]
decomposition
group
of
a
closed
point
x
of
X
as
the
order
of
x.
We
shall
refer
to
the
[manifestly
finite!]
least
common
multiple
of
the
orders
of
the
closed
points
of
X
as
the
order
of
X.
Thus,
it
follows
immediately
from
the
definitions
that
X
is
a
hyperbolic
curve
if
and
only
if
the
order
of
X
is
equal
to
1.
6
SHINICHI
MOCHIZUKI
Section
1:
Maximal
Abelian
Cuspidalizations
Let
X
be
a
proper
hyperbolic
curve
over
a
field
k
which
is
either
finite
or
nonarchimedean
local.
Write
d
k
for
the
cohomological
dimension
of
k.
Thus,
if
k
is
finite
(respectively,
nonar-
chimedean
local),
then
d
k
=
1
(respectively,
d
k
=
2
[cf.,
e.g.,
[NSW],
Chapter
7,
Theorem
7.1.8,
(i)]).
If
k
is
finite
(respectively,
nonarchimedean
local),
we
shall
denote
the
characteristic
of
k
(respectively,
of
the
residue
field
of
k)
by
p
and
the
number
p
(respectively,
1)
by
p
†
.
Also,
we
shall
write
Primes
†
=
Primes\(Primes
def
{p
†
})
[where
Primes
is
the
set
of
all
prime
numbers
[cf.
§0];
the
intersection
is
taken
in
the
“ambient
set”
Z].
Let
Σ
be
a
set
of
prime
numbers
that
contains
at
least
one
prime
number
that
is
invertible
in
k.
Write
Σ
=
Σ\(Σ
def
{p});
Σ
†
=
Σ\(Σ
def
{p
†
})
the
max-
[where
the
intersections
are
taken
in
the
“ambient
set”
Z].
Denote
by
Z
and
by
Z
†
the
maximal
pro-Σ
†
quotient
of
Z.
imal
pro-Σ
quotient
of
Z
If
k
is
an
algebraic
closure
of
k,
then
we
shall
denote
the
result
of
base-changing
objects
over
k
to
k
by
means
of
a
subscript
“k”.
Any
choice
of
a
basepoint
of
X
determines
an
algebraic
closure
k
of
k,
and
hence
an
exact
sequence
1
→
π
1
(X
k
)
→
π
1
(X)
→
G
k
→
1
def
where
G
k
=
Gal(k/k);
π
1
(X),
π
1
(X
k
)
are
the
étale
fundamental
groups
of
X,
def
X
k
,
respectively.
Write
Δ
X
for
the
maximal
pro-Σ
quotient
of
π
1
(X
k
)
and
Π
X
=
π
1
(X)/Ker(π
1
(X
k
)
Δ
X
).
Thus,
we
have
an
exact
sequence:
1
→
Δ
X
→
Π
X
→
G
k
→
1
def
Similarly,
if
we
write
X
×
X
=
X
×
k
X,
then
we
obtain
[by
considering
the
maximal
pro-Σ
quotient
of
π
1
((X
×
X)
k
)]
an
exact
sequence
1
→
Δ
X×X
→
Π
X×X
→
G
k
→
1
where
Π
X×X
(respectively,
Δ
X×X
)
may
be
identified
with
Π
X
×
G
k
Π
X
(respectively,
Δ
X
×
Δ
X
).
Let
Π
Z
⊆
Π
X×X
be
an
open
subgroup
that
surjects
onto
G
k
.
Write
def
Z
→
X
×
X
for
the
corresponding
covering;
Δ
Z
=
Ker(Π
Z
G
k
).
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
7
†
A
Proposition
1.1.
(Group-theoreticity
of
Étale
Cohomology)
Let
Z
be
a
finite
quotient,
and
N
a
finite
A-module
equipped
with
a
continuous
Δ
X
-
(respectively,
Π
X
-;
Δ
Z
-;
Π
Z
-)
action.
Then
for
i
∈
Z,
the
natural
homomorphism
i
H
i
(Δ
X
,
N
)
→
H
ét
(X
k
,
N
)
i
(respectively,
H
i
(Π
X
,
N
)
→
H
ét
(X,
N
);
i
(Z
k
,
N
);
H
i
(Δ
Z
,
N
)
→
H
ét
i
H
i
(Π
Z
,
N
)
→
H
ét
(Z,
N
))
is
an
isomorphism.
Proof.
In
light
of
the
Leray
spectral
sequence
for
the
surjections
Π
X
G
k
,
Π
Z
Im(Π
Z
)
⊆
Π
X
[i.e.,
where
“Im(−)”
denotes
the
image
via
the
natural
homomorphism
associated
to
one
of
the
projections
Z
→
X
×
X
→
X],
it
suffices
to
verify
the
asserted
isomorphism
in
the
case
of
Δ
X
.
If
Y
→
X
k
is
a
connected
finite
étale
Galois
Σ-covering,
then
the
associated
Leray
spectral
sequence
has
“E
2
-term”
given
by
the
cohomology
of
Gal(Y
/X)
with
coefficients
in
the
étale
cohomology
of
Y
and
abuts
to
the
étale
cohomology
of
X
k
.
By
allowing
Y
to
vary,
one
then
verifies
immediately
that
it
suffices
to
verify
that
every
étale
cohomology
class
of
Y
[with
coefficients
in
N
]
vanishes
upon
pull-back
to
some
[connected]
finite
étale
Σ-covering
Y
→
Y
.
Moreover,
by
passing
to
an
appropriate
Y
,
one
reduces
immediately
to
the
case
where
N
=
A,
equipped
with
the
trivial
Π
X
-action.
Then
the
vanishing
assertion
in
question
is
a
tautology
for
“H
1
”;
for
“H
2
”,
it
suffices
to
take
Y
→
Y
so
that
the
degree
of
Y
→
Y
annihilates
A
[cf.,
e.g.,
the
discussion
at
the
bottom
of
[FK],
p.
136].
Set:
†
),
Z
†
);
M
X
=
Hom
Z
†
(H
2
(Δ
X
,
Z
def
def
⊗d
−1
M
k
=
Hom
Z
†
(H
d
k
(G
k
,
M
X
k
⊗d
−1
),
M
X
k
)
†
-modules
of
rank
one;
M
X
is
isomorphic
as
a
G
k
-module
Thus,
M
k
,
M
X
are
free
Z
†
(1)
via
the
†
(1)
[where
the
“(1)”
denotes
a
“Tate
twist”
—
i.e.,
G
k
acts
on
Z
to
Z
†
(d
−1).
[Indeed,
this
cyclotomic
character];
M
k
is
isomorphic
as
a
G
k
-module
to
Z
k
follows
from
Proposition
1.1;
Poincaré
duality
[cf.,
e.g.,
[FK],
Chapter
II,
Theorem
[together
with
an
easy
compu-
1.13];
the
fact,
in
the
finite
field
case,
that
G
k
∼
=
Z
the
well-known
theory
of
the
cohomology
of
tation
of
the
group
cohomology
of
Z];
nonarchimedean
local
fields
[cf.,
e.g.,
[NSW],
Chapter
7,
Theorem
7.2.6].]
Remark
1.2.0.
Note
that
for
any
open
subgroup
Π
X
⊆
Π
X
[which
we
think
of
as
corresponding
to
a
finite
étale
covering
X
→
X],
we
obtain
a
natural
isomorphism
∼
M
X
→
M
X
†
)
to
the
induced
morphism
on
group
coho-
by
applying
the
functor
Hom
Z
†
(−,
Z
†
)
→
H
2
(Δ
X
,
Z
†
)
[where
Δ
X
def
=
Ker(Π
X
→
G
k
)]
and
dividing
mology
H
2
(Δ
X
,
Z
by
[Δ
X
:
Δ
X
].
[One
verifies
easily
that
this
does
indeed
yield
an
isomorphism
—
cf.,
e.g.,
the
discussion
at
the
bottom
of
[FK],
p.
136.]
Moreover,
relative
to
8
SHINICHI
MOCHIZUKI
†
,
H
2
(Δ
X
,
M
X
)
∼
†
,
the
iso-
the
tautological
isomorphisms
H
2
(Δ
X
,
M
X
)
∼
=
Z
=
Z
∼
morphism
M
X
→
M
X
just
constructed
induces
[via
the
restriction
morphism
on
†
given
by
multiplication
by
[Δ
X
:
Δ
X
].
†
→
Z
group
cohomology]
the
morphism
Z
Similarly,
if
k
is
the
base
field
of
X
,
then
we
obtain
a
natural
isomorphism
∼
M
k
→
M
k
∼
by
applying
the
natural
isomorphism
M
X
→
M
X
just
constructed
and
the
dual
of
the
natural
pull-back
morphism
on
group
cohomology
and
then
dividing
by
[k
:
k]
[cf.,
e.g.,
[NSW],
Chapter
7,
Corollary
7.1.4].
Proposition
1.2.
(Top
Cohomology
Modules)
(i)
There
are
natural
isomorphisms:
†
;
H
d
k
(G
k
,
M
k
)
∼
=
Z
†
;
H
2
(Δ
X
,
M
X
)
∼
=
Z
⊗2
∼
†
H
4
(Δ
Z
,
M
X
)
=
Z
;
†
H
d
k
+2
(Π
X
,
M
X
⊗
M
k
)
∼
=
Z
⊗2
†
H
d
k
+4
(Π
Z
,
M
X
⊗
M
k
)
∼
=
Z
∼
†
(ii)
There
is
a
unique
isomorphism
M
X
→
Z
(1)
such
that
the
image
of
†
†
∼
1
∈
Z
maps
via
the
composite
of
the
isomorphism
Z
=
H
2
(Δ
X
,
M
X
)
of
(i)
∼
†
(1))
induced
by
the
isomorphism
with
the
isomorphism
H
2
(Δ
X
,
M
X
)
→
H
2
(Δ
X
,
Z
∼
†
M
X
→
Z
(1)
in
question
to
the
[first]
Chern
class
of
a
line
bundle
of
degree
1
on
X
k
.
Proof.
Assertion
(i)
follows
from
the
definitions;
the
Leray
spectral
sequence
for
the
surjections
Π
X
G
k
,
Π
Z
Im(Π
Z
)
⊆
Π
X
[i.e.,
where
“Im(−)”
denotes
the
image
via
the
natural
homomorphism
associated
to
one
of
the
projections
Z
→
X
×
X
→
X].
Assertion
(ii)
is
immediate
from
the
definitions.
†
A
be
a
finite
quotient,
and
Proposition
1.3.
(Duality)
For
i
∈
Z,
let
Z
N
a
finite
A-module.
(i)
Suppose
that
N
is
equipped
with
a
continuous
G
k
-action.
Then
the
pairing
H
i
(G
k
,
N
)
×
H
d
k
−i
(G
k
,
Hom
A
(N,
M
k
⊗
A))
→
A
determined
by
the
cup
product
in
group
cohomology
and
the
natural
isomorphisms
of
Proposition
1.2,
(i),
determines
an
isomorphism
as
follows:
∼
H
i
(G
k
,
N
)
→
Hom
A
(H
d
k
−i
(G
k
,
Hom
A
(N,
M
k
⊗
A)),
A)
(ii)
Suppose
that
N
is
equipped
with
a
continuous
Π
X
-
(respectively,
Δ
X
-;
Π
Z
-;
Δ
Z
-)
action.
Then
the
pairing
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
9
H
i
(Π
X
,
N
)
×
H
d
k
+2−i
(Π
X
,
Hom
A
(N,
M
X
⊗
M
k
⊗
A))
→
A
(respectively,
H
i
(Δ
X
,
N
)
×
H
2−i
(Δ
X
,
Hom
A
(N,
M
X
⊗
A))
→
A;
⊗2
⊗
M
k
⊗
A))
→
A;
H
i
(Π
Z
,
N
)
×
H
d
k
+4−i
(Π
Z
,
Hom
A
(N,
M
X
⊗2
H
i
(Δ
Z
,
N
)
×
H
4−i
(Δ
Z
,
Hom
A
(N,
M
X
⊗
A))
→
A)
determined
by
the
cup
product
in
group
cohomology
and
the
natural
isomorphisms
of
Proposition
1.2,
(i),
determines
an
isomorphism
as
follows:
∼
H
i
(Π
X
,
N
)
→
Hom
A
(H
d
k
+2−i
(Π
X
,
Hom
A
(N,
M
X
⊗
M
k
⊗
A)),
A)
∼
(respectively,
H
i
(Δ
X
,
N
)
→
Hom
A
(H
2−i
(Δ
X
,
Hom
A
(N,
M
X
⊗
A)),
A);
∼
⊗2
⊗
M
k
⊗
A)),
A);
H
i
(Π
Z
,
N
)
→
Hom
A
(H
d
k
+4−i
(Π
Z
,
Hom
A
(N,
M
X
∼
⊗2
⊗
A)),
A))
H
i
(Δ
Z
,
N
)
→
Hom
A
(H
4−i
(Δ
Z
,
Hom
A
(N,
M
X
[together
with
Proof.
Assertion
(i)
follows
immediately
from
the
fact
that
G
k
∼
=
Z
in
the
finite
field
case;
[NSW],
an
easy
computation
of
the
group
cohomology
of
Z]
Chapter
7,
Theorem
7.2.6,
in
the
nonarchimedean
local
field
case.
Assertion
(ii)
then
follows
from
assertion
(i);
the
Leray
spectral
sequences
associated
to
Π
X
G
k
,
Π
Z
Im(Π
Z
)
⊆
Π
X
[i.e.,
where
“Im(−)”
denotes
the
image
via
the
natural
homomorphism
associated
to
one
of
the
projections
Z
→
X
×
X
→
X];
Proposition
1.1;
Poincaré
duality
[cf.,
e.g.,
[FK],
Chapter
II,
Theorem
1.13].
Proposition
1.4.
(Automorphisms
of
Cyclotomic
Extensions)
(i)
We
have:
H
0
(G
k
,
H
1
(Δ
X
,
M
X
))
=
0.
(ii)
There
are
natural
isomorphisms
∼
∼
∧
∼
∼
∧
H
1
(Π
X
,
M
X
)
→
H
1
(G
k
,
M
X
)
→
(k
×
)
H
1
(Π
Z
,
M
X
)
→
H
1
(G
k
,
M
X
)
→
(k
×
)
—
where
the
first
isomorphisms
in
each
line
are
induced
by
the
surjections
Π
X
G
k
,
Π
Z
G
k
;
the
second
isomorphisms
in
each
line
are
induced
by
the
isomor-
∧
phism
of
Proposition
1.2,
(ii),
and
the
Kummer
exact
sequence;
(k
×
)
is
the
max-
imal
pro-Σ
†
-quotient
of
k
×
.
Proof.
Assertion
(i)
follows
immediately
from
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
[cf.,
e.g.,
[Mumf],
p.
206]
in
the
finite
field
case;
[Mzk8],
Lemma
4.6,
in
the
nonarchimedean
local
field
case.
The
first
isomorphisms
of
assertion
(ii)
follow
immediately
from
assertion
(i)
and
the
Leray
spectral
sequences
10
SHINICHI
MOCHIZUKI
associated
to
Π
X
G
k
,
Π
Z
G
k
;
the
second
isomorphisms
follow
immediately
from
consideration
of
the
Kummer
exact
sequence
for
Spec(k).
Definition
1.5.
(i)
Let
H
be
a
profinite
group
equipped
with
a
homomorphism
H
→
Π
X
.
Then
we
shall
refer
to
the
kernel
I
H
of
H
→
Π
X
as
the
cuspidal
subgroup
of
H
[relative
to
H
→
Π
X
].
We
shall
say
that
H
is
cuspidally
abelian
(respectively,
cuspidally
pro-Σ
∗
[where
Σ
∗
is
a
set
of
prime
numbers])
[relative
to
H
→
Π
X
]
if
I
H
is
abelian
(respectively,
a
pro-Σ
∗
group).
If
H
is
cuspidally
abelian,
then
observe
that
H/I
H
acts
naturally
[by
conjugation]
on
I
H
;
we
shall
say
that
H
is
cuspidally
central
[relative
to
H
→
Π
X
]
if
this
action
of
H/I
H
on
I
H
is
trivial.
Also,
we
shall
use
similar
terminology
to
the
terminology
just
introduced
for
H
→
Π
X
when
Π
X
is
replaced
by
Δ
X
,
Π
X×X
,
Δ
X×X
.
(ii)
Let
H
be
a
profinite
group;
H
1
⊆
H
a
closed
subgroup.
Then
we
shall
refer
to
as
an
H
1
-inner
automorphism
of
H
an
inner
automorphism
induced
by
conjugation
by
an
element
of
H
1
.
If
H
is
also
a
profinite
group,
then
we
shall
refer
to
as
an
H
1
-outer
homomorphism
H
→
H
an
equivalence
class
of
homo-
morphisms
H
→
H,
where
two
such
homomorphisms
are
considered
equivalent
if
they
differ
by
composition
by
an
H
1
-inner
automorphism.
If
H
is
equipped
with
a
homomorphism
H
→
G
k
[cf.,
e.g.,
the
various
groups
introduced
above],
and
def
H
1
=
Ker(H
→
G
k
),
then
we
shall
refer
to
an
H
1
-inner
automorphism
(respec-
tively,
H
1
-outer
homomorphism)
as
a
geometrically
inner
automorphism
(respec-
tively,
geometrically
outer
homomorphism).
If
H
is
equipped
with
a
structure
of
extension
of
some
other
profinite
group
H
0
by
a
finite
product
H
1
of
copies
of
M
X
,
or,
more
generally,
a
projective
limit
H
1
of
such
finite
products,
then
we
shall
refer
to
an
H
1
-inner
automorphism
(respectively,
H
1
-outer
homomorphism)
as
a
cyclo-
tomically
inner
automorphism
(respectively,
cyclotomically
outer
homomorphism).
If
H
is
equipped
with
a
homomorphism
to
Π
X
,
Δ
X
,
Π
X×X
,
or
Δ
X×X
[cf.
the
situation
of
(i)],
and
H
1
is
the
kernel
of
this
homomorphism,
then
we
shall
refer
to
an
H
1
-inner
automorphism
(respectively,
H
1
-outer
homomorphism)
as
a
cuspidally
inner
automorphism
(respectively,
cuspidally
outer
homomorphism).
Next,
let
Π
X
⊆
Π
X
be
an
open
normal
subgroup,
corresponding
to
a
finite
étale
Galois
covering
X
→
X.
Set
def
Π
Z
=
Π
X
×X
·
Π
X
⊆
Π
X×X
[where
we
regard
Π
X
as
a
subgroup
of
Π
X×X
via
the
diagonal
map];
write
Z
→
X
×
X
for
the
covering
determined
by
Π
Z
.
Thus,
it
is
a
tautology
that
the
diagonal
morphism
ι
:
X
→
X
×
X
lifts
to
a
morphism
ι
:
X
→
Z
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
11
which
induces
the
inclusion
Π
X
→
Π
Z
on
fundamental
groups.
If
Z
→
X
×
X
is
a
connected
finite
étale
covering
arising
from
an
open
subgroup
of
Π
X×X
,
write:
def
U
X×X
=
(X
×
X)\ι(X);
def
U
Z
=
(U
X×X
)
×
(X×X)
Z
Denote
by
Δ
U
X×X
the
maximal
cuspidally
[i.e.,
relative
to
the
natural
map
to
π
1
((X
×
X)
k
)]
pro-Σ
†
quotient
of
the
maximal
pro-Σ
quotient
of
the
tame
funda-
mental
group
of
(U
X×X
)
k
[where
“tame”
is
with
respect
to
the
divisor
ι(X)
⊆
X
×
X]
and
by
Π
U
X×X
the
quotient
π
1
(U
X×X
)/Ker(π
1
((U
X×X
)
k
)
Δ
U
X×X
);
write
Π
U
Z
⊆
Π
U
X×X
for
the
open
subgroup
corresponding
to
the
finite
étale
cover-
ing
U
Z
→
U
X×X
.
Proposition
1.6.
(Characteristic
Class
of
the
Diagonal)
(i)
The
pull-back
morphism
arising
from
the
natural
inclusion
Π
X
→
Π
Z
(⊆
Π
X×X
=
Π
X
×
G
k
Π
X
)
composed
with
the
natural
isomorphism
of
Proposition
1.2,
(i),
determines
a
homo-
morphism
∼
†
H
d
k
+2
(Π
Z
,
M
X
⊗
M
k
)
→
H
d
k
+2
(Π
X
,
M
X
⊗
M
k
)
→
Z
hence
[by
Proposition
1.3,
(ii)]
a
class
diag
∈
H
2
(Π
Z
,
M
X
)
η
Z
which
is
equal
to
the
étale
cohomology
class
associated
to
ι
(X)
⊆
Z
,
or,
alterna-
tively,
the
[first]
Chern
class
of
the
line
bundle
O
Z
(ι
(X)).
(ii)
Denote
by
L
×
diag
[Z
]
→
Z
the
complement
of
the
zero
section
in
the
geometric
line
bundle
[i.e.,
G
m
-torsor]
determined
by
O
Z
(ι
(X)),
by
Δ
L
×
[Z
]
the
maximal
cuspidally
pro-Σ
†
quotient
of
diag
the
maximal
pro-Σ
quotient
of
the
tame
fundamental
group
of
(L
×
diag
[Z
])
k
[where
“tame”
is
with
respect
to
the
divisor
determined
by
the
complement
of
the
G
m
-
1
torsor
L
×
diag
[Z
]
in
the
naturally
associated
P
-bundle],
and
by
Π
L
×
[Z
]
the
quotient
diag
×
π
1
(L
×
).
Then
[in
light
of
the
isomorphism
diag
[Z
])/Ker(π
1
((L
diag
[Z
])
k
)
Δ
L
×
diag
[Z
]
of
Proposition
1.2,
(ii)]
we
have
a
natural
exact
sequence
1
→
M
X
→
Π
L
×
diag
[Z
]
→
Π
Z
→
1
diag
whose
associated
extension
class
is
equal
to
the
class
η
Z
.
(iii)
The
global
section
of
O
Z
(ι
(X))
over
Z
determined
by
the
natural
inclu-
sion
O
Z
→
O
Z
(ι
(X))
defines
a
morphism
U
Z
→
L
×
diag
[Z
]
12
SHINICHI
MOCHIZUKI
over
Z
which
induces
a
surjective
homomorphism
of
groups
over
Π
Z
:
Π
U
Z
Π
L
×
diag
[Z
]
Proof.
Assertion
(i)
follows
immediately
from
Propositions
1.1,
1.2,
1.3,
together
with
well-known
facts
concerning
Chern
classes
and
associated
cycles
in
étale
co-
homology
[cf.,
e.g.,
[FK],
Chapter
II,
Definition
1.2,
Proposition
2.2].
Assertion
(ii)
follows
from
Proposition
1.1;
[Mzk7],
Definition
4.2,
Lemmas
4.4,
4.5.
Asser-
tion
(iii)
follows
from
[Mzk8],
Lemma
4.2,
by
considering
fibers
over
one
of
the
two
natural
projections
Π
Z
→
Π
X×X
Π
X
.
[Here,
we
note
that
although
in
[Mzk7],
§4;
[Mzk8],
the
base
field
is
assumed
to
be
of
characteristic
zero,
one
ver-
ifies
immediately
that
the
same
arguments
as
those
applied
in
loc.
cit.
yield
the
corresponding
results
in
the
finite
field
case
—
so
long
as
we
restrict
the
coefficients
†
.]
of
the
cohomology
modules
in
question
to
modules
over
Z
Definition
1.7.
(i)
We
shall
refer
to
a
covering
Z
→
X
×
X
as
in
the
above
discussion
as
the
diagonal
covering
associated
to
the
covering
X
→
X.
We
shall
refer
to
an
extension
of
profinite
groups
1
→
M
X
→
D
→
Π
Z
→
1
diag
whose
associated
extension
class
is
the
class
η
Z
of
Proposition
1.6,
(i),
as
a
fun-
damental
extension
[of
Π
Z
].
In
the
following
(ii)
—
(iv),
we
shall
assume
that
1
→
M
X
→
D
→
Π
X×X
→
1
is
a
fundamental
extension.
(ii)
Let
x,
y
∈
X(k);
write
D
x
,
D
y
⊆
Π
X
for
the
associated
decomposition
groups
[which
are
well-defined
up
to
conjugation
by
an
element
of
Δ
X
—
cf.
Remark
1.7.1
below].
Now
set:
def
D
x
=
D|
D
x
×
Gk
Π
X
;
def
D
x,y
=
D|
D
x
×
Gk
D
y
Thus,
D
x
(respectively,
D
x,y
)
is
an
extension
of
Π
X
(respectively,
G
k
)
by
M
X
.
Similarly,
if
D
=
i
m
i
·
x
i
,
E
=
j
n
j
·
y
j
are
divisors
on
X
supported
on
points
that
are
rational
over
k,
then
set:
def
D
D
=
i
m
i
·
D
x
i
;
def
D
D,E
=
m
i
·
n
j
·
D
x
i
,y
j
i,j
[where
the
sums
are
to
be
understood
as
sums
of
extensions
of
Π
X
or
G
k
by
M
X
—
i.e.,
the
sums
are
induced
by
the
additive
structure
of
M
X
].
Also,
we
shall
write
def
C
=
−D|
Π
X
[where
we
regard
Π
X
as
a
subgroup
of
Π
X×X
via
the
diagonal
map].
[Thus,
C
is
an
extension
of
Π
X
by
M
X
whose
extension
class
is
the
Chern
class
of
the
canonical
bundle
of
X.]
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
13
(iii)
Let
S
⊆
X(k)
be
a
finite
subset.
Then
we
shall
write
def
D
S
=
D
x
x∈S
[where
the
product
is
to
be
understood
as
the
fiber
product
over
Π
X
].
Thus,
D
S
is
an
extension
of
Π
X
by
a
product
of
copies
of
M
X
indexed
by
elements
of
S.
We
shall
refer
to
D
S
as
a
maximal
abelian
S-cuspidalization
[of
Π
X
at
S].
Observe
that
if
T
⊆
X(k)
is
a
finite
subset
such
that
S
⊆
T
,
then
we
obtain
a
natural
projection
morphism
D
T
→
D
S
.
(iv)
We
shall
refer
to
a
homomorphism
Π
U
X×X
→
D
over
Π
X×X
as
a
fundamental
section
if,
for
some
isomorphism
of
D
with
Π
L
×
diag
that
induces
the
identity
on
Π
X×X
,
M
X
,
the
resulting
composite
homomorphism
Π
U
X×X
→
Π
L
×
is
the
homomorphism
of
Proposition
1.6,
(iii).
diag
Remark
1.7.1.
Relative
to
the
situation
in
Definition
1.7,
(ii),
conjugation
by
elements
δ
∈
Δ
X
induces
isomorphisms
between
the
different
possible
choices
of
“D
x
”,
all
of
which
lie
over
the
isomorphism
between
any
of
these
choices
and
G
k
induced
by
the
projection
Π
X
G
k
.
Moreover,
by
lifting
(δ,
1)
∈
Δ
X×X
⊆
Π
X×X
to
an
element
δ
D
∈
D,
and
conjugating
by
δ
D
,
we
obtain
natural
isomorphisms
between
the
various
resulting
“D
x
’s”
which
induce
the
identity
on
the
quotient
group
D
x
Π
X
,
as
well
as
on
the
subgroup
M
X
⊆
D
x
.
Note
that
this
last
property
[i.e.,
of
inducing
the
identity
on
Π
X
,
M
X
]
holds
precisely
because
we
are
working
with
δ
∈
Δ
X
⊆
Π
X
,
as
opposed
to
an
arbitrary
“δ
∈
Π
X
”.
Remark
1.7.2.
By
Proposition
1.4,
(ii),
if
E
is
any
profinite
group
extension
of
Π
X
(respectively,
G
k
;
an
open
subgroup
Π
Z
⊆
Π
X×X
that
surjects
onto
G
k
)
by
M
X
,
then
the
group
of
cyclotomically
outer
automorphisms
of
the
extension
E
[i.e.,
that
induce
the
identity
on
Π
X
(respectively,
G
k
;
Π
Z
)
and
M
X
]
may
be
naturally
∧
identified
with
(k
×
)
.
In
particular,
in
the
context
of
Definition
1.7,
(iv),
any
two
fundamental
sections
of
D
differ,
up
to
composition
with
a
cyclotomically
inner
∧
automorphism
of
D,
by
a
“(k
×
)
-multiple”.
Next,
if
k
is
nonarchimedean
local,
then
set
G
†
k
=
G
k
;
if
k
is
finite,
then
write
†
].
Also,
we
shall
use
G
†
k
⊆
G
k
for
the
maximal
pro-Σ
†
subgroup
of
G
k
[so
G
†
k
∼
=
Z
the
notation
def
Π
†
(−)
=
Π
(−)
×
G
k
G
†
k
⊆
Π
(−)
def
[where
“(−)”
is
any
smooth,
geometrically
connected
scheme
over
k,
with
arithmetic
fundamental
group
Π
(−)
G
k
].
14
SHINICHI
MOCHIZUKI
Proposition
1.8.
tions)
Let
(Basic
Properties
of
Maximal
Abelian
Cuspidaliza-
1
→
M
X
→
D
→
Π
X×X
→
1
be
a
fundamental
extension;
φ
:
Π
U
X×X
D
a
fundamental
section;
S
⊆
X(k)
a
finite
subset.
Then:
(i)
The
profinite
groups
Δ
X×X
,
Δ
X
,
as
well
as
any
profinite
group
extension
or
Π
†
X
by
a
[possibly
empty]
finite
product
of
copies
of
M
X
is
slim
[cf.
of
Π
†
X×X
§0].
In
particular,
the
profinite
group
D
S
†
=
D
S
×
G
k
G
†
k
is
slim.
def
def
(ii)
For
x
∈
X(k),
write
U
x
=
X\{x}.
Denote
by
Δ
U
x
the
maximal
cuspidally
[i.e.,
relative
to
the
natural
map
to
π
1
((U
x
)
k
)]
pro-Σ
†
quotient
of
the
maximal
pro-Σ
quotient
of
the
tame
fundamental
group
of
(U
x
)
k
[where
“tame”
is
with
respect
to
the
complement
of
U
x
in
X]
and
by
Π
U
x
the
quotient
π
1
(U
x
)/Ker(π
1
((U
x
)
k
)
Δ
U
x
).
Then
the
inverse
image
via
either
of
the
natural
projections
Π
U
X×X
Π
X
of
the
decomposition
group
D
x
⊆
Π
X
is
naturally
isomorphic
to
Π
U
x
.
In
particular,
Δ
U
X×X
,
Π
†
U
X×X
are
slim.
(iii)
For
S
⊆
X(k)
a
finite
subset,
write:
def
U
S
=
U
x
x∈S
[where
the
product
is
to
be
understood
as
the
fiber
product
over
X].
Denote
by
Δ
U
S
the
maximal
cuspidally
[i.e.,
relative
to
the
natural
map
to
π
1
((U
S
)
k
)]
pro-Σ
†
quotient
of
the
maximal
pro-Σ
quotient
of
the
tame
fundamental
group
of
(U
S
)
k
[where
“tame”
is
with
respect
to
the
complement
of
U
S
in
X],
and
by
Π
U
S
the
quotient
π
1
(U
S
)/Ker(π
1
((U
S
)
k
)
Δ
U
S
).
Then
Δ
U
S
,
Π
†
U
S
are
slim.
Forming
the
product
of
the
specializations
of
φ
to
the
various
D
x
×
G
k
Π
X
⊆
Π
X×X
yields
homomorphisms
Π
U
x
→
D
S
Π
U
S
→
x∈S
[where
the
product
is
to
be
understood
as
the
fiber
product
over
Π
X
].
Moreover,
the
def
composite
morphism
Π
U
S
→
D
S
is
surjective;
the
resulting
quotient
of
Δ
U
S
=
Ker(Π
U
S
G
k
)
is
the
maximal
cuspidally
central
quotient
of
Δ
U
S
,
relative
to
the
surjection
Δ
U
S
Δ
X
.
def
(iv)
The
quotient
of
Δ
U
X×X
=
Ker(Π
U
X×X
G
k
)
determined
by
φ
:
Π
U
X×X
D
is
the
maximal
cuspidally
central
quotient
of
Δ
U
X×X
,
relative
to
the
sur-
jection
Δ
U
X×X
Δ
X×X
.
Proof.
Assertion
(i)
follows
immediately
from
the
slimness
of
Π
†
X
,
Δ
X
[cf.,
e.g.,
[Mzk5],
Theorem
1.1.1,
(ii);
the
proofs
of
[Mzk5],
Lemmas
1.3.1,
1.3.10],
together
with
the
[easily
verified]
fact
that
G
†
k
acts
faithfully
on
M
X
via
the
cyclotomic
char-
acter.
Next,
we
consider
assertion
(ii).
The
portion
of
assertion
(ii)
concerning
Π
U
x
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
15
follows
immediately
from
the
existence
of
the
“homotopy
exact
sequence
associated
to
a
family
of
curves”
[cf.,
e.g.,
[Stix],
Proposition
2.3].
The
slimness
assertion
then
follows
from
assertion
(i)
[applied
to
Π
†
X
]
and
the
slimness
of
Δ
U
x
[cf.
the
proofs
of
[Mzk5],
Lemmas
1.3.1,
1.3.10].
As
for
assertion
(iii),
the
slimness
of
Δ
U
S
,
Π
†
U
S
follows
via
the
arguments
given
in
the
proofs
of
[Mzk5],
Lemmas
1.3.1,
1.3.10.
The
existence
of
homomorphisms
Π
U
S
→
x∈S
Π
U
x
→
D
S
as
asserted
is
immediate
from
the
definitions,
assertion
(ii).
For
x
∈
S,
write
D
x
[U
S
]
⊆
Π
U
S
for
the
decomposition
group
of
x;
I
x
[U
S
]
⊆
D
x
[U
S
]
for
the
inertia
subgroup.
Now
it
is
immediate
from
the
definitions
that
I
x
[U
S
]
maps
isomorphically
onto
the
copy
M
X
in
D
S
corresponding
to
the
point
x.
This
implies
the
desired
surjectivity.
Since,
moreover,
it
is
immediate
from
the
definitions
that
the
cuspidal
subgroup
of
any
cuspidally
central
quotient
of
Δ
U
S
is
generated
by
the
image
of
the
I
x
[U
S
],
as
x
ranges
over
the
elements
of
S,
the
final
assertion
concerning
the
maximal
cuspidally
central
quotient
of
Δ
U
S
follows
immediately.
Assertion
(iv)
follows
by
a
similar
argument
to
the
argument
applied
to
the
final
portion
of
assertion
(iii).
Next,
let
Z
→
X
×
X
(respectively,
Z
→
X
×
X;
Z
∗
→
X
×
X)
be
the
diagonal
covering
associated
to
a
covering
X
→
X
(respectively,
X
→
X;
X
∗
→
X)
arising
from
an
open
subgroup
of
Π
X
;
denote
by
ι
:
X
→
Z
(respectively,
ι
:
X
→
Z
;
ι
∗
:
X
→
Z
∗
)
the
tautological
lifting
of
the
diagonal
embedding
ι
:
X
→
X
×
X
and
by
k
(respectively,
k
;
k
∗
)
the
extension
of
k
determined
by
X
(respectively,
X
;
X
∗
).
Assume,
moreover,
that
the
covering
X
→
X
factors
as
follows:
X
→
X
→
X
∗
→
X
Thus,
we
obtain
a
factorization
Z
→
Z
→
Z
∗
→
X
×
X.
Let
1
→
M
X
→
D
→
Π
Z
→
1
be
a
fundamental
extension
of
Π
Z
.
Write
1
→
M
X
→
D
X
×X
→
Π
X
×X
→
1
for
the
pull-back
of
the
extension
D
via
the
inclusion
Π
X
×X
⊆
Π
Z
.
Now
if
we
think
of
Π
X×X
or
Π
X
×X
as
only
being
defined
up
to
Δ
X
×
{1}-inner
automorphisms,
then
it
makes
sense,
for
δ
∈
Δ
X
/Δ
X
to
speak
of
the
pull-back
of
the
extension
D
X
×X
via
δ
×
1:
1
→
M
X
→
(δ
×
1)
∗
D
X
×X
→
Π
X
×X
→
1
In
particular,
we
may
form
the
fiber
product
over
Π
X
×X
:
def
S
X
/X
∗
(D
)
X
×X
=
δ∈Δ
X
∗
/Δ
X
(δ
×
1)
∗
D
X
×X
16
SHINICHI
MOCHIZUKI
Thus,
S
X
/X
∗
(D
)
X
×X
is
an
extension
of
Π
X
×X
by
a
product
of
copies
of
M
X
indexed
by
Δ
X
∗
/Δ
X
;
S
X
/X
∗
(D
)
X
×X
admits
a
tautological
Δ
X
×
{1}-
outer
[more
precisely:
a
(Δ
X
×
{1})
×
Π
X
×X
S
X
/X
∗
(D
)
X
×X
-outer]
action
by
the
finite
group
Δ
X
∗
/Δ
X
∼
=
(Δ
X
∗
/Δ
X
)
×
{1}.
Moreover,
the
natural
outer
∼
action
of
Gal(X
/X)
=
Gal((X
×
X
)/Z
)
∼
=
Π
X
/Π
X
on
Π
X
×X
[arising
from
the
diagonal
embedding
Π
X
→
Π
Z
]
clearly
lifts
to
an
outer
action
of
Gal(X
/X)
on
S
X
/X
∗
(D
)
X
×X
,
which
is
compatible,
relative
to
the
natural
ac-
tion
of
Gal(X
/X)
on
Δ
X
∗
/Δ
X
by
conjugation,
with
the
Δ
X
×
{1}-outer
action
of
Δ
X
∗
/Δ
X
on
S
X
/X
∗
(D
)
X
×X
.
Thus,
in
summary,
the
natural
isomorphism
(Δ
X
∗
/Δ
X
)
×
{1}
Gal(X
/X)
∼
=
Gal((X
×
X
)/Z
∗
)
determines
a
homomorphism
Gal((X
×
X
)/Z
∗
)
→
Out(S
X
/X
∗
(D
)
X
×X
)
via
which
we
may
pull-back
the
extension
“1
→
(−)
→
Aut(−)
→
Out(−)
→
1”
[cf.
§0;
Proposition
1.8,
(i)]
for
S
X
/X
∗
(D
)
X
×X
to
obtain
an
extension
1
→
M
X
→
S
X
/X
∗
(D
)
→
Π
Z
∗
→
1
Δ
X
∗
/Δ
X
in
which
Π
Z
∗
is
only
determined
up
to
Δ
X
×
{1}-inner
automorphisms.
Note,
moreover,
that
every
cyclotomically
outer
automorphism
of
the
extension
D
—
∧
i.e.,
an
element
of
(k
×
)
[cf.
Remark
1.7.2]
—
induces
a
cyclotomically
outer
automorphism
of
S
X
/X
∗
(D
).
In
particular,
we
have
a
natural
cyclotomically
outer
∧
action
of
(k
×
)
on
S
X
/X
∗
(D
).
Next,
let
us
push-forward
the
extension
S
X
/X
∗
(D
)
just
constructed
via
the
natural
surjection
M
X
M
X
Δ
X
∗
/Δ
X
Δ
X
∗
/Δ
X
[which
induces
the
identity
morphism
M
X
→
M
X
between
the
various
factors
of
the
domain
and
codomain],
so
as
to
obtain
an
extension
Tr
X
/X
:X
∗
(D
)
as
follows:
M
X
→
Tr
X
/X
:X
∗
(D
)
→
Π
Z
∗
→
1
1
→
Δ
X
∗
/Δ
X
[in
which
Π
Z
∗
is
only
determined
up
to
Δ
X
×
{1}-inner
automorphisms].
Proposition
1.9.
discussion
above:
(Symmetrizations
and
Traces)
In
the
notation
of
the
(i)
The
extension
Tr
X
/X
:X
(D
)
of
Π
Z
by
M
X
is
a
fundamental
exten-
sion
of
Π
Z
.
(ii)
There
is
a
natural
commutative
diagram:
1
−→
M
X
−→
S
X
/X
(D
)
Δ
X
/Δ
X
1
−→
Δ
X
/Δ
X
M
X
−→
S
X
/X
(Tr
X
/X
:X
(D
))
Π
X×X
−→
1
−→
1
⏐
⏐
⏐
⏐
⏐
⏐
−→
id
−→
Π
X×X
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
17
[which
is
well-defined
up
to
Δ
X
×
{1}-inner
automorphisms
—
cf.
Remark
1.9.1
below].
(iii)
Relative
to
the
commutative
diagram
of
(ii),
the
natural
cyclotomically
∧
∧
outer
action
of
(k
×
)
on
S
X
/X
(D
)
lies
over
the
composite
of
the
map
(k
×
)
→
∧
(k
×
)
given
by
raising
to
the
[Δ
X
:
Δ
X
]-power
with
the
natural
cyclotomically
∧
outer
action
of
(k
×
)
on
S
X
/X
(Tr
X
/X
:X
(D
)).
In
particular,
if
N
is
a
positive
integer
that
divides
[Δ
X
:
Δ
X
],
then
the
natural
cyclotomically
outer
action
of
∧
an
element
of
(k
×
)
on
S
X
/X
(D
)
lies
over
the
cyclotomically
outer
action
of
an
∧
element
of
{(k
×
)
}
N
on
S
X
/X
(Tr
X
/X
:X
(D
)).
Proof.
that
To
verify
assertion
(i),
observe
that
it
is
immediate
from
the
definitions
ι
(X)
×
Z
(X
×
X
)
⊆
X
×
X
is
equal
to
the
Δ
X
/Δ
X
-orbit
of
ι
(X)×
Z
(X
×
X
)
⊆
X
×
X
.
Now
assertion
(i)
follows
by
translating
this
observation
into
the
language
of
étale
cohomology
classes
associated
to
subvarieties;
assertions
(ii),
(iii)
follow
formally
from
assertion
(i)
and
the
definitions
of
the
various
objects
involved.
Remark
1.9.1.
Relative
to
the
commutative
diagram
of
Proposition
1.9,
(ii),
note
that,
although
S
X
/X
(Tr
X
/X
:X
(D
))
is,
by
definition,
only
well-defined
up
to
Δ
X
×
{1}-inner
automorphisms,
the
push-forward
of
S
X
/X
(D
)
by
M
X
→
M
X
Δ
X
/Δ
X
Δ
X
/Δ
X
is
well-defined
up
to
Δ
X
×
{1}-inner
automorphisms.
That
is
to
say,
the
push-
forward
extension
implicit
in
this
commutative
diagram
furnishes
a
canonically
more
rigid
version
of
the
extension
S
X
/X
(Tr
X
/X
:X
(D
)).
Definition
1.10.
(i)
We
shall
refer
to
the
extension
S
X
/X
∗
(D
)
[of
Π
Z
∗
]
constructed
from
the
fundamental
extension
D
as
the
[X
/X
∗
-]symmetrization
of
D
,
or,
alter-
natively,
as
a
symmetrized
fundamental
extension.
We
shall
refer
to
the
extension
Tr
X
/X
:X
∗
(D
)
[of
Π
Z
∗
]
constructed
from
the
fundamental
extension
D
as
the
[X
/X
:
X
∗
-]trace
of
D
,
or,
alternatively,
as
a
trace-symmetrized
fundamental
extension.
(ii)
If
D
is
a
fundamental
extension
of
Π
Z
,
then
we
shall
refer
to
as
a
morphism
of
trace
type
any
morphism
S
X
/X
(D
)
→
S
X
/X
(D
)
obtained
by
composing
the
morphism
S
X
/X
(D
)
→
S
X
/X
(Tr
X
/X
:X
(D
))
18
SHINICHI
MOCHIZUKI
of
Proposition
1.9,
(ii),
with
a
morphism
S
X
/X
(Tr
X
/X
:X
(D
))
→
S
X
/X
(D
)
arising
[by
the
functoriality
of
the
construction
of
“S
X
/X
(−)”]
from
an
isomorphism
∼
of
[fundamental]
extensions
Tr
X
/X
:X
(D
)
→
D
of
Π
Z
by
M
X
[which
induces
the
identity
on
Π
Z
,
M
X
].
(iii)
We
shall
refer
to
as
a
pro-symmetrized
fundamental
extension
any
com-
patible
system
[indexed
by
the
natural
numbers]
.
.
.
S
i
.
.
.
S
j
.
.
.
Π
X×X
of
morphisms
of
trace
type
[up
to
inner
automorphisms
of
the
appropriate
type]
be-
tween
symmetrized
fundamental
extensions,
where
S
i
is
the
X
i
/X-symmetrization
of
a
fundamental
extension
of
Π
Z
i
;
Z
i
is
the
diagonal
covering
associated
to
an
open
normal
subgroup
Π
X
i
⊆
Π
X
;
the
intersection
of
the
Π
X
i
is
trivial.
In
this
situation,
we
shall
refer
to
the
inverse
limit
profinite
group
def
lim
S
∞
=
←
−
S
i
i
as
the
limit
of
the
pro-symmetrized
fundamental
extension
{S
i
};
any
profinite
group
S
∞
arising
in
this
fashion
will
be
referred
to
as
a
pro-fundamental
extension
[of
Π
X×X
].
(iv)
Let
S
⊆
X(k)
be
a
finite
subset;
S
an
X
/X-symmetrization
of
a
funda-
mental
extension
D
of
Π
Z
.
Then
we
shall
write
def
S
S
=
S
D
x
×
G
Π
X
k
x∈S
[where
the
product
is
to
be
understood
as
the
fiber
product
over
Π
X
].
Thus,
S
S
is
an
extension
of
Π
X
by
a
product
of
copies
of
M
X
.
Similarly,
given
a
projective
system
{S
i
}
as
in
(iii),
we
obtain
a
projective
system
{(S
i
)
S
},
with
inverse
limit:
(S
∞
)
S
We
shall
refer
to
(S
∞
)
S
as
a
maximal
abelian
S-pro-cuspidalization
[of
Π
X
at
S].
Observe
that
if
T
⊆
X(k)
is
a
finite
subset
such
that
S
⊆
T
,
then
we
obtain
a
natural
projection
morphism
(S
∞
)
T
→
(S
∞
)
S
.
Remark
1.10.1.
Let
D
be
as
in
Definition
1.7,
(iii);
S
,
{S
i
},
S
∞
as
in
Definition
1.10,
(iii),
(iv).
Then
observe
that
it
follows
from
Proposition
1.8,
(i),
that
the
†
[i.e.,
the
result
of
applying
“×
G
k
G
†
k
”
to
“daggered
versions”
D
†
,
(S
)
†
,
S
i
†
,
and
S
∞
D,
S
,
S
i
,
and
S
∞
]
are
slim.
In
particular,
if
S
⊆
X
cl
is
any
finite
set
of
closed
points
of
X,
then
we
may
form
the
objects
D
S
†
;
(S
)
†
S
;
(S
i
)
†
S
;
(S
∞
)
†
S
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
19
by
passing
to
a
Galois
covering
X
k
S
→
X
[i.e.,
the
result
of
base-changing
X
to
some
finite
Galois
extension
k
S
of
k]
such
that
the
closed
points
of
X
k
S
that
lie
over
points
of
S
are
rational
over
k
S
;
forming
the
various
objects
in
question
over
X
k
S
[cf.
Definition
1.7,
(iii);
Definition
1.10,
(iv)];
and,
finally,
“descending
to
X”
via
the
natural
outer
action
of
G
k
/G
†
k
S
on
the
various
objects
in
question
[cf.
the
exact
sequence
“1
→
(−)
→
Aut(−)
→
Out(−)
→
1”
of
§0;
the
slimness
mentioned
above].
Thus,
in
the
remainder
of
this
paper,
we
shall
often
speak
of
the
various
objects
defined
in
Definition
1.7,
(iii);
Definition
1.10,
(iv),
even
when
the
points
of
the
finite
set
S
are
not
necessarily
rational
over
k.
Before
proceeding,
we
note
the
following:
Lemma
1.11.
(Conjugacy
Estimate)
Let
H
⊆
Δ
X
be
a
normal
open
subgroup;
a
∈
Δ
X
/H
an
element
not
equal
to
the
identity;
N
a
Σ
†
-integer
[cf.
§0].
Then
there
exists
a
normal
open
subgroup
H
⊆
Δ
X
contained
in
H
such
that
for
any
normal
open
subgroup
H
⊆
Δ
X
contained
in
H
and
any
a
∈
Δ
X
/H
that
lifts
a,
the
cardinality
of
the
H-conjugacy
class
Conj(a
,
H
)
⊆
Δ
X
/H
of
a
in
Δ
X
/H
is
divisible
by
N
.
Proof.
In
the
notation
of
the
statement
of
Lemma
1.11,
denote
by
Z(a
,
H
)
⊆
H
the
subgroup
of
elements
δ
∈
H
such
that
δ
·
a
·
δ
−1
=
a
in
Δ
X
/H
.
Then
it
is
immediate
that
if
a
is
the
image
of
a
in
Δ
X
/H
,
then
Z(a
,
H
)
⊆
Z(a
,
H
),
so
the
cardinality
of
Conj(a
,
H
)
∼
=
H/Z(a
,
H
)
is
divisible
by
the
cardinality
∼
of
Conj(a
,
H
)
=
H/Z(a
,
H
).
Thus,
it
suffices
to
find
a
normal
open
subgroup
H
⊆
H
such
that
for
any
a
∈
Δ
X
/H
that
lifts
a,
the
cardinality
of
Conj(a
,
H
)
is
divisible
by
N
.
To
this
end,
let
us
consider,
for
some
prime
number
l
∈
Σ
†
,
the
maximal
pro-l
quotient
H[l]
of
the
abelianization
H
ab
of
H.
Note
that
Δ
X
/H
acts
by
conjugation
on
H
ab
,
H[l].
Now
I
claim
that
there
exists
a
[nonzero]
h
l
∈
H[l]
such
that
a(h
l
)
=
h
l
.
Indeed,
if
this
claim
were
false,
then
it
would
follow
that
a
acts
trivially
on
H[l].
But
since
a
induces
a
nontrivial
automorphism
of
the
covering
of
X
k
determined
by
H,
it
follows
that
a
induces
a
nontrivial
automorphism
of
the
l-power
torsion
points
of
the
Jacobian
of
X
k
[since
these
points
are
Zariski
dense
in
this
Jacobian]
—
a
contradiction.
This
completes
the
proof
of
the
claim.
Now
let
j
∈
H
be
an
element
that
lifts
the
various
h
l
obtained
above
for
the
[finite
collection
of]
primes
l
that
divide
N
;
let
a
X
∈
Δ
X
be
an
element
that
lifts
a.
Then
observe
that
for
some
integer
power
M
of
N
that
is
independent
of
the
ab
⊗
(Z/M
Z)
is
nonzero,
for
n
∈
Z
choice
of
a
X
,
the
image
of
j
n
·
a
X
·
j
−n
·
a
−1
X
in
H
Thus,
if
we
take
H
equal
to
the
inverse
image
of
with
nonzero
image
in
Z/N
·
Z.
M
·
H
ab
in
H(⊆
Δ
X
),
we
obtain
that
the
intersection
of
the
subgroup
j
Z
⊆
H
with
with
nonzero
Z(a
,
H
)
[where
a
∈
Δ
X
/H
lifts
a]
does
not
contain
j
n
,
for
n
∈
Z
Z
But
this
implies
that
the
intersection
(j
)
Z(a
,
H
)
⊆
j
N·
Z
,
image
in
Z/N
·
Z.
hence
that
[H
:
Z(a
,
H
)]
is
divisible
by
N
,
as
desired.
20
SHINICHI
MOCHIZUKI
Next,
we
consider
the
following
fundamental
extensions
of
Π
Z
,
Π
Z
:
D
=
Π
L
×
def
diag
[Z
]
D
=
Tr
X
/X
:X
(D
)
def
;
[cf.
Proposition
1.6,
(ii)].
Note
that
in
this
situation,
it
follows
immediately
from
∼
the
definitions
that
we
obtain
a
natural
isomorphism
D
→
Π
L
×
[Z
]
,
which
we
shall
diag
use
in
the
following
discussion
to
identify
D
,
Π
L
×
[Z
]
.
Thus,
we
have
fundamental
diag
sections:
Π
U
Z
D
;
Π
U
Z
D
[cf.
Proposition
1.6,
(iii)].
In
particular,
by
pulling
back
from
Z
to
X
×
X
,
we
obtain
a
surjection:
Π
U
X
×X
D
X
×X
Now
if
we
apply
the
natural
outer
(Δ
X
/Δ
X
)
×
{1}-action
on
Π
U
X
×X
to
this
surjection,
it
follows
from
the
definition
of
“S
X
/X
(D
)”
that
we
obtain
a
natural
homomorphism
Π
U
X
×X
S
X
/X
(D
)
X
×X
which
is
easily
verified
[cf.
Proposition
1.8,
(ii),
(iii)]
to
be
surjective.
Since,
more-
over,
the
construction
of
this
surjective
homomorphism
is
manifestly
compatible
with
the
outer
actions
of
Gal(X
/X)
on
both
sides,
we
thus
obtain
a
natural
sur-
jection:
Π
U
X×X
S
X
/X
(D
)
Now
let
us
denote
by
D
X
⊆
Π
U
X×X
the
decomposition
group
of
the
subvariety
ι(X)
⊆
X
×
X.
[Thus,
D
X
is
well-defined
up
to
conjugation;
here,
we
assume
that
we
have
chosen
a
conjugate
that
maps
to
the
image
of
the
diagonal
embedding
Π
X
→
Π
X×X
via
the
natural
surjection
Π
U
X×X
Π
X×X
.]
Observe
that
we
have
a
natural
exact
sequence
1
→
I
X
→
D
X
→
Π
X
→
1
[where
I
X
—
i.e.,
the
inertia
subgroup
of
D
X
—
is
defined
so
as
to
make
the
sequence
exact],
together
with
a
natural
isomorphism
I
X
∼
=
M
X
.
Also,
we
shall
def
def
write
D
X
=
D
X
Π
U
X
×X
;
D
X
=
D
X
Π
U
X
×X
.
Since
the
construction
just
carried
out
for
double
primed
objects
may
also
be
carried
out
for
single
primed
objects,
we
thus
obtain
the
following:
Proposition
1.12.
(Symmetrized
Fundamental
Sections)
In
the
notation
of
the
discussion
above:
(i)
There
is
a
natural
commutative
diagram:
D
X
⏐
⏐
⊆
D
X
⊆
id
Π
U
X×X
⏐
⏐
S
X
/X
(D
)
⏐
⏐
Π
U
X×X
S
X
/X
(D
)
id
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
21
[where
the
vertical
arrow
on
the
right
is
the
morphism
in
the
diagram
of
Proposition
1.9,
(ii)].
(ii)
Denote
by
means
of
a
subscript
X
the
result
of
pulling
back
extensions
of
Π
X×X
,
Π
Z
,
Π
X
×X
to
Π
X
[via
the
diagonal
inclusion].
Then
the
projection
[cf.
the
fiber
product
defining
S
X
/X
(D
)]
to
the
factor
labeled
“Δ
X
/Δ
X
”
detemines
a
natural
surjection
ζ
:
S
X
/X
(D
)
X
D
X
whose
restriction
to
D
X
[i.e.,
relative
to
the
arrows
in
the
first
line
of
the
com-
∼
mutative
diagram
of
(i)]
defines
an
isomorphism
D
X
→
D
X
.
Moreover,
the
cuspidal
subgroup
of
D
X
maps
isomorphically
onto
the
factor
of
M
X
in
S
X
/X
(D
)
labeled
“Δ
X
/Δ
X
”.
In
particular,
if
we
denote
by
S
X
/X
(D
)
=
the
quotient
of
S
X
/X
(D
)
by
this
factor
of
M
X
,
then
ζ
determines
a
surjection
ζ
=
:
S
X
/X
(D
)
=
X
Π
X
∼
whose
restriction
to
the
quotient
D
X
Π
X
is
equal
to
the
identity
Π
X
→
Π
X
[up
to
geometric
inner
automorphisms].
Thus,
we
have
a
natural
commutative
diagram
[well-defined
up
to
geometric
inner
automorphisms]
ζ
D
X
⏐
⏐
→
S
X
/X
(D
)
X
⏐
⏐
−→
Π
X
→
S
X
/X
(D
)
=
X
−→
D
X
⏐
⏐
ζ
=
Π
X
in
which
the
two
horizontal
composites
are
isomorphisms;
the
vertical
arrows
are
surjections;
both
squares
are
cartesian.
(iii)
If
we
carry
out
the
construction
of
(ii)
for
the
single
primed
objects,
then
the
commutative
diagram
of
(i)
induces
a
natural
commutative
diagram
[well-
defined
up
to
geometric
inner
automorphisms]:
Π
X
⏐
⏐
Π
X
→
S
X
/X
(D
)
=
X
⏐
⏐
→
S
X
/X
(D
)
=
X
ζ
=
−→
Π
X
⏐
⏐
ζ
=
−→
Π
X
Moreover,
there
is
a
natural
outer
action
of
Gal(X
/X)
(respectively,
Gal(X
/X))
on
the
first
(respectively,
second)
line
of
this
diagram;
these
outer
actions
are
com-
patible
with
one
another.
(iv)
When
considered
up
to
cyclotomically
inner
automorphisms,
the
sections
form
a
torsor
over
the
group
of
ζ
=
(Δ
X
/Δ
X
)\(Δ
X
/Δ
X
)
∧
((k
)
×
)
22
SHINICHI
MOCHIZUKI
[where
the
“\”
denotes
the
set-theoretic
complement].
The
Gal(X
/X)-equivariant
form
a
torsor
over
the
Gal(X
/X)-invariant
subgroup
of
this
group.
sections
of
ζ
=
Similar
statements
hold
for
the
single
primed
objects.
(v)
The
double
and
single
primed
torsors
of
equivariant
sections
of
(iv)
are
related,
via
the
right-hand
square
of
the
diagram
of
(iii),
by
a
homomorphism
((k
)
×
)
∧
(Δ
X
/Δ
X
)
\(Δ
X
/Δ
X
)
Gal(X
/X)
→
((k
)
×
)
∧
Gal(X
/X)
(Δ
X
/Δ
X
)
\(Δ
X
/Δ
X
)
[where
the
superscripts
denote
the
result
of
taking
invariants
with
respect
to
the
action
of
the
superscripted
group]
that
satisfies
the
following
property:
An
element
ξ
of
the
domain
maps
to
an
element
of
the
codomain
whose
component
in
the
factor
labeled
a
∈
Δ
X
/Δ
X
is
a
product
of
elements
of
∧
((k
)
×
)
of
the
form
N
k
/k
(λ
)
n
.
a
∧
Here,
a
∈
(Δ
X
/Δ
X
)\(Δ
X
/Δ
X
)
maps
to
a
in
Δ
X
/Δ
X
;
λ
∈
((k
)
×
)
is
the
component
of
ξ
in
the
factor
labeled
a
;
k
a
is
an
intermediate
field
extension
∧
∧
∧
between
k
and
k
such
that
λ
∈
((k
a
)
×
)
;
N
k
a
/k
:
((k
a
)
×
)
→
((k
)
×
)
is
the
norm
map;
n
is
the
cardinality
of
the
Δ
X
-conjugacy
class
of
a
in
(Δ
X
/Δ
X
).
In
particular,
by
Lemma
1.11
[where
we
take
“H”
to
be
Δ
X
,
“H
”
to
be
Δ
X
],
for
a
given
Δ
X
,
if,
for
a
given
positive
integer
N
,
Δ
X
is
“sufficiently
small”,
then
an
arbitrary
Gal(X
/X)-equivariant
section
of
ζ
=
lies
over
the
canonical
section
of
ζ
=
given
in
(iii),
up
to
the
cyclotomically
outer
action
of
some
N
-th
power
of
an
element
of
the
single
primed
version
of
the
group
exhibited
in
the
display
of
(iv).
Proof.
All
of
these
assertions
follow
immediately
from
the
definitions
[and,
in
the
case
of
assertion
(iv),
Proposition
1.4,
(ii)].
Definition
1.13.
Let
D
be
a
fundamental
extension
of
Π
Z
;
{S
i
}
a
pro-
symmetrized
fundamental
extension,
with
limit
S
∞
[cf.
Definition
1.10,
(iii)].
(i)
We
shall
refer
to
as
a
symmetrized
fundamental
section
a
homomorphism
Π
U
X×X
S
X
/X
(D
)
obtained
by
composing
the
surjection
Π
U
X×X
S
X
/X
(D
)
of
Proposition
1.12,
∼
(i),
with
the
isomorphism
S
X
/X
(D
)
→
S
X
/X
(D
)
induced
by
an
isomorphism
∼
D
→
D
of
fundamental
extensions
of
Π
Z
by
M
X
[which
induces
the
identity
on
Π
Z
,
M
X
].
We
shall
refer
to
an
inclusion
D
X
→
S
X
/X
(D
)
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
23
obtained
by
restricting
a
symmetrized
fundamental
section
to
D
X
⊆
Π
U
X×X
[cf.
Proposition
1.12,
(i)]
as
a
fundamental
inclusion.
(ii)
We
shall
refer
to
a
compatible
system
of
symmetrized
fundamental
sections
Π
U
X×X
→
S
i
as
a
pro-symmetrized
fundamental
section
and
to
the
resulting
limit
homomorphism
Π
U
X×X
→
S
∞
as
a
pro-fundamental
section.
Similarly,
we
have
a
notion
of
“pro-fundamental
inclusions”.
Remark
1.13.1.
Thus,
by
the
above
discussion,
if
we
take
the
“S
i
”
to
be
the
symmetrizations
of
the
Π
L
×
[Z
]
as
in
Proposition
1.6,
(ii),
then
we
obtain
natural
diag
pro-fundamental
sections
and
pro-fundamental
inclusions
[cf.
Proposition
1.12,
(i),
(ii),
(iii)].
Proposition
1.14.
(Maximal
Cuspidally
Abelian
Quotients)
Let
{S
i
}
be
a
pro-symmetrized
fundamental
extension,
with
limit
S
∞
[cf.
Definition
1.10,
(iii)]
and
pro-fundamental
section
Π
U
X×X
S
∞
[cf.
Definition
1.13,
(ii)];
S
⊆
X
cl
a
finite
set
of
closed
points
[cf.
Remark
1.10.1].
Then:
(i)
The
pro-fundamental
section
Π
U
X×X
S
∞
determines
a
surjection
Π
U
S
(S
∞
)
S
[cf.
Proposition
1.8,
(iii)].
The
resulting
quotient
of
Δ
U
S
(respectively,
Π
U
S
)
is
the
maximal
cuspidally
abelian
quotient
Δ
U
S
Δ
c-ab
U
S
(respectively,
Π
U
S
c-ab
Π
U
S
)
of
Δ
U
S
(respectively,
Π
U
S
).
(ii)
The
quotient
of
Δ
U
X×X
(respectively,
Π
U
X×X
)
induced
by
the
pro-funda-
mental
section
Π
U
X×X
S
∞
is
the
maximal
cuspidally
abelian
quotient
c-ab
[which
we
shall
denote
by]
Δ
U
X×X
Δ
c-ab
U
X×X
(respectively,
Π
U
X×X
Π
U
X×X
)
of
Δ
U
X×X
(respectively,
Π
U
X×X
).
Proof.
Indeed,
this
follows
as
in
the
proof
of
Proposition
1.8,
(iii),
(iv),
by
observ-
ing
that
the
cuspidal
subgroup
of
the
maximal
cuspidally
abelian
quotient
of
Δ
U
S
(respectively,
Δ
U
X×X
)
is
naturally
isomorphic
to
the
inverse
limit
of
the
cuspidal
subgroups
of
the
maximal
cuspidally
central
quotients
of
the
Δ
U
S
×
Δ
X
Δ
X
(⊆
Δ
U
S
)
(respectively,
Δ
U
X
×X
)
[as
Δ
X
⊆
Δ
X
ranges
over
the
open
normal
subgroups
of
Δ
X
].
Proposition
1.15.
(Automorphisms
and
Commensurators)
Let
{S
i
}
be
a
pro-symmetrized
fundamental
extension,
with
limit
S
∞
[cf.
Definition
1.10,
(iii)]
and
pro-fundamental
inclusion
D
X
→
S
∞
[cf.
Definition
1.13,
(ii)].
Then:
(i)
Any
automorphism
α
of
the
profinite
group
Π
c-ab
U
X×X
which
(a)
is
compatible
with
the
natural
surjection
Π
c-ab
U
X×X
Π
X×X
and
induces
the
identity
on
Π
X×X
;
24
SHINICHI
MOCHIZUKI
(b)
preserves
the
image
of
M
X
∼
=
I
X
⊆
D
X
via
the
natural
inclusion
D
X
→
Π
c-ab
U
X×X
is
cuspidally
inner.
(ii)
Π
X
(respectively,
Δ
X
)
is
commensurably
terminal
[cf.
§0]
in
Π
X×X
(respectively,
Δ
X×X
).
(iii)
D
X
is
commensurably
terminal
in
S
i
,
S
∞
∼
=
Π
c-ab
U
X×X
.
Proof.
First,
we
verify
assertion
(i).
By
Proposition
1.14,
(ii),
we
have
a
natural
∼
isomorphism
Π
c-ab
U
X×X
→
S
∞
,
so
we
may
think
of
α
as
an
automorphism
of
S
∞
.
In
light
of
(a);
Proposition
1.8,
(iii),
it
follows
that
α
is
compatible
with
the
natural
surjections
S
∞
S
i
.
Write
α
i
for
the
automorphism
of
S
i
induced
by
α.
By
(a),
(b),
it
follows
that
α
i
is
an
automorphism
of
the
extension
S
i
of
Π
X×X
by
a
product
of
copies
of
M
X
which
induces
the
identity
on
both
Π
X×X
and
the
product
of
copies
of
M
X
[cf.
the
definition
by
a
certain
fiber
product
of
the
symmetrized
fundamental
extension
S
i
].
[Here,
we
note
that
the
fact
that
α
i
induces
the
identity
on
each
copy
of
M
X
follows
by
considering
the
non-torsion
[cf.
Propositions
1.2,
(ii);
1.6,
(i),
(ii)]
extension
class
determined
by
that
copy
of
M
X
[which
is
preserved
by
α
i
!],
together
with
the
fact
that
α
i
induces
the
identity
on
the
second
cohomology
groups
of
open
subgroups
of
Δ
X×X
with
coefficients
in
M
X
.]
Thus,
up
to
cyclotomically
inner
∧
automorphisms,
α
i
arises
from
a
collection
of
elements
of
(k
i
×
)
,
where
k
i
is
some
finite
Galois
extension
of
k
[cf.
Proposition
1.4,
(ii)],
one
corresponding
to
each
copy
of
M
X
.
Moreover,
since
these
copies
of
M
X
are
permuted
by
the
action
of
Π
X×X
by
conjugation,
it
follows
that
[up
to
cyclotomically
inner
automorphisms]
∧
∧
∧
α
i
arises
from
a
single
element
of
(k
i
×
)
,
which
in
fact
belongs
to
(k
×
)
(⊆
(k
i
×
)
)
[as
one
sees
by
considering
the
conjugation
action
via
the
“G
k
portion”
of
Π
X×X
].
On
the
other
hand,
since
the
α
i
form
a
compatible
system
of
automorphisms
of
the
∧
S
i
,
it
follows
from
Proposition
1.9,
(iii),
that
this
element
of
(k
×
)
must
be
equal
to
1,
as
desired.
Next,
to
verify
assertion
(ii),
let
us
observe
that
it
suffices
to
show
that
Δ
X
is
commensurably
terminal
in
Δ
X×X
.
But
this
follows
immediately
from
the
fact
that
Δ
X
is
slim
[cf.
Proposition
1.8,
(i)].
Finally,
we
consider
assertion
(iii).
Clearly,
it
suffices
to
show
that
D
X
is
commensurably
terminal
in
S
i
.
By
assertion
(ii),
to
verify
this
commensurable
terminality,
it
suffices
to
show
that
the
[manifestly
abelian]
cuspidal
subgroup
H
i
⊆
S
i
[i.e.,
relative
to
the
natural
surjection
S
i
Π
X×X
]
satisfies
the
following
property:
Every
h
∈
H
i
such
that
h
δ
−
h
∈
D
X
,
for
all
δ
in
some
open
subgroup
J
of
D
X
,
satisfies
h
∈
D
X
.
But
this
property
follows
immediately
[cf.
the
definition
by
a
certain
fiber
product
of
the
symmetrized
fundamental
extension
S
i
]
from
the
fact
that,
for
J
sufficiently
small,
the
J-module
H
i
/(D
X
H
i
)
is
isomorphic
to
a
direct
product
of
a
finite
number
of
copies
of
M
X
.
The
following
result
is
the
main
result
of
the
present
§1:
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
25
Theorem
1.16.
(Reconstruction
of
Maximal
Cuspidally
Abelian
Quo-
tients)
Let
X,
Y
be
hyperbolic
curves
over
a
finite
or
nonarchimedean
local
field;
denote
the
base
fields
of
X,
Y
by
k
X
,
k
Y
,
respectively.
Let
Σ
X
(respectively,
Σ
Y
)
be
a
set
of
prime
numbers
that
contains
at
least
one
prime
number
that
is
invertible
in
k
X
(respectively,
k
Y
);
write
Δ
X
(respectively,
Δ
Y
)
for
the
maximal
cuspidally
pro-Σ
†
X
(respectively,
pro-Σ
†
Y
)
quotient
of
the
maximal
pro-Σ
X
(respectively,
pro-Σ
Y
)
quotient
of
the
tame
fundamental
group
of
X
k
X
(re-
spectively,
Y
k
Y
)
[where
“tame”
is
with
respect
to
the
complement
of
X
k
X
(respec-
tively,
Y
k
Y
)
in
its
canonical
compactification],
and
Π
X
(respectively,
Π
Y
)
for
the
corresponding
quotient
of
the
étale
fundamental
group
of
X
(respectively,
Y
).
Let
∼
α
:
Π
X
→
Π
Y
be
an
isomorphism
of
profinite
groups.
Then:
(i)
We
have
Σ
†
X
=
Σ
†
Y
;
write
Σ
†
=
Σ
†
X
=
Σ
†
Y
.
Moreover,
k
X
is
a
finite
field
if
and
only
if
k
Y
is;
α
preserves
the
decomposition
groups
of
cusps;
X
is
of
type
(g,
r)
[where
g,
r
≥
0
are
integers
such
that
2g
−
2
+
r
>
0]
if
and
only
if
Y
is
of
type
(g,
r).
Finally,
if
k
X
,
k
Y
are
nonarchimedean
local,
then
their
residue
characteristics
coincide.
def
(ii)
α
is
compatible
with
the
natural
quotients
Π
X
G
k
X
,
Π
Y
G
k
Y
.
(iii)
Assume
that
X,
Y
are
proper.
Denote
by
Π
U
X×X
Π
c-ab
U
X×X
,
Π
U
Y
×Y
c-ab
Π
U
Y
×Y
the
maximal
cuspidally
[i.e.,
relative
to
the
natural
surjections
Π
U
X×X
Π
X×X
,
Π
U
Y
×Y
Π
Y
×Y
]
abelian
quotients
[cf.
Proposition
1.14].
Then
there
is
a
commutative
diagram
[well-defined
up
to
cuspidally
inner
automorphisms]
Π
c-ab
U
X×X
⏐
⏐
Π
X×X
α
c-ab
−→
Π
c-ab
U
Y
×Y
⏐
⏐
α×α
−→
Π
Y
×Y
—
where,
the
horizontal
arrows
are
isomorphisms
which
are
compatible
with
c-ab
the
natural
inclusions
D
X
→
Π
c-ab
U
X×X
,
D
Y
→
Π
U
Y
×Y
[cf.
Proposition
1.12,
(i)];
the
vertical
arrows
are
the
natural
surjections.
Finally,
the
correspondence
α
→
α
c-ab
is
functorial
[up
to
cuspidally
inner
automorphisms]
with
respect
to
α.
Proof.
First,
we
consider
assertions
(i),
(ii).
Note
that
k
X
is
finite
if
and
only
if,
for
every
open
subgroup
H
⊆
Π
X
,
the
quotient
of
the
abelianization
H
ab
by
the
closure
of
the
torsion
subgroup
of
H
ab
is
topologically
cyclic
[cf.
[Tama],
Proposition
3.3,
(ii)];
a
similar
statement
holds
for
k
Y
,
Π
Y
.
Thus,
k
X
is
finite
if
and
only
if
k
Y
is.
Now
suppose
that
k
X
,
k
Y
are
finite.
Then
assertion
(ii)
also
follows
from
[Tama],
26
SHINICHI
MOCHIZUKI
Proposition
3.3,
(ii).
The
fact
that
Σ
†
X
=
Σ
†
Y
then
follows
from
the
following
observation:
The
subset
Σ
†
X
⊆
Primes
is
the
subset
on
which
the
function
Primes
l
→
dim
Q
l
((Δ
X
)
ab
⊗
Q
l
)
attains
its
maximum
value
[cf.
[Tama],
Proposition
3.1];
a
similar
statement
holds
for
Y
.
Now
by
considering
the
respective
outer
actions
of
G
k
X
,
G
k
Y
on
the
max-
imal
pro-l
quotients
of
Δ
X
,
Δ
Y
,
for
some
l
∈
Σ
†
,
we
obtain
that
α
preserves
the
decomposition
groups
of
cusps
[hence
that
X
is
of
type
(g,
r)
if
and
only
if
Y
is
of
type
(g,
r)],
by
[Mzk9],
Corollary
2.7,
(i).
This
completes
the
proof
of
assertions
(i),
(ii)
in
the
finite
field
case.
Next,
let
us
assume
that
k
X
,
k
Y
are
nonarchimedean
local.
Then
the
portion
of
assertion
(i)
concerning
Σ
X
=
Σ
†
X
,
Σ
Y
=
Σ
†
Y
follows
by
considering
the
cohomo-
logical
dimension
of
Π
X
,
Π
Y
—
cf.,
e.g.,
Proposition
1.3,
(ii)
[in
the
proper
case].
def
As
for
assertion
(ii),
if
the
cardinality
of
Σ
=
Σ
†
is
≥
2,
then
assertion
(ii)
follows
from
the
evident
pro-Σ
analogue
of
[Mzk5],
Lemma
1.3.8;
if
the
cardinality
of
Σ
is
1,
then
assertion
(ii)
follows
from
Lemma
1.17,
(c),
(d)
below.
Now
the
portion
of
assertion
(i)
concerning
the
residue
characteristics
of
k
X
,
k
Y
follows
from
assertion
(ii)
and
[Mzk5],
Proposition
1.2.1,
(i);
the
fact
that
α
preserves
the
decomposition
groups
of
cusps
[hence
that
X
is
of
type
(g,
r)
if
and
only
if
Y
is
of
type
(g,
r)]
follows
from
[Mzk9],
Corollary
2.7,
(i).
This
completes
the
proof
of
assertions
(i),
(ii)
in
the
nonarchimedean
local
field
case.
Finally,
we
consider
assertion
(iii).
It
follows
from
the
definitions
that
α
induces
∼
→
X,
Z
Y
→
Y
are
diagonal
an
isomorphism
M
X
→
M
Y
.
If,
moreover,
Z
X
coverings
corresponding
to
[connected]
finite
étale
Galois
coverings
X
→
X,
Y
→
Y
that
arise
from
open
subgroups
of
Π
X
,
Π
Y
that
correspond
via
α,
then
α
induces
an
isomorphism
of
group
cohomology
modules
∼
2
,
M
X
)
→
H
(Π
Z
,
M
Y
)
H
2
(Π
Z
X
Y
,
that
preserves
the
extension
classes
associated
to
fundamental
extensions
of
Π
Z
X
Π
Z
Y
[cf.
Proposition
1.6,
(i)].
In
particular,
if
D
(respectively,
E
)
is
a
fundamental
(respectively,
Π
Z
Y
),
then
α
induces
an
isomorphism
extension
of
Π
Z
X
∼
D
→
E
∼
∼
→
Π
Z
already
induced
which
is
compatible
with
the
morphisms
M
X
→
M
Y
,
Π
Z
X
Y
by
α,
and,
moreover,
uniquely
determined,
up
to
cyclotomically
inner
automor-
∧
×
∧
phisms,
and
the
action
of
(k
X
)
(respectively,
(k
Y
×
)
)
[cf.
Proposition
1.4,
(ii)].
On
the
other
hand,
by
allowing
X
,
Y
to
vary,
taking
symmetrizations
of
the
fundamen-
tal
extensions
involved
[which
may
be
constructed
entirely
group-theoretically!],
and
making
use
of
the
vertical
morphism
in
the
center
of
the
diagram
of
Proposition
1.9,
(ii)
[again
an
object
which
may
be
constructed
entirely
group-theoretically!],
it
follows
from
Proposition
1.9,
(iii),
that
the
indeterminacy
of
the
isomorphism
∧
∼
×
∧
D
→
E
arising
from
the
action
of
(k
X
)
,
(k
Y
×
)
“converges
to
the
identity
inde-
∼
terminacy”
[i.e.,
by
taking
D
→
E
to
arise
as
just
described
from
an
isomorphism
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
27
∼
of
fundamental
extensions
D
→
E
associated
to
[connected]
finite
étale
coverings
X
→
X
,
Y
→
Y
[that
arise
from
open
subgroups
of
Π
X
,
Π
Y
that
correspond
via
α],
where
the
open
subgroups
Π
X
⊆
Π
X
,
Π
Y
⊆
Π
Y
are
sufficiently
small].
Thus,
in
light
of
the
manifest
functoriality
of
the
vertical
morphism
in
the
center
of
the
diagram
of
Proposition
1.9,
(ii)
[the
detailed
explication
of
which,
in
terms
of
various
commutative
diagrams,
is
a
routine
task
which
we
leave
to
the
reader!],
we
obtain
an
isomorphism
∼
{S
i
}
→
{T
j
}
of
pro-symmetrized
fundamental
extensions
[cf.
Definition
1.10,
(iii)]
of
Π
X×X
,
Π
Y
×Y
,
respectively,
which
arises
from
α
and
is
completely
determined
up
to
cyclo-
tomically
inner
automorphisms.
Here,
we
pause
to
note
that
although
in
the
con-
struction
of
the
symmetrization
of
a
fundamental
extension
D
(respectively,
E
),
one
must,
a
priori,
contend
with
a
certain
indeterminacy
with
respect
to
Δ
X
×{1}-
(respectively,
Δ
Y
×
{1}-)inner
automorphisms
[cf.,
e.g.,
Proposition
1.9,
(ii)],
in
fact,
by
allowing
X
,
Y
to
vary,
this
indeterminacy
also
“converges
to
the
identity
indeterminacy”
[cf.
Remark
1.9.1].
Thus,
in
summary,
α
induces
an
isomorphism
[well-defined
up
to
cyclotomically
[or,
alternatively,
cuspidally]
inner
automorphisms]
∼
S
∞
→
T
∞
of
pro-fundamental
extensions
of
Π
X×X
,
Π
Y
×Y
,
respectively.
Moreover,
by
apply-
ing
the
fact
that
the
left-hand
square
of
the
commutative
diagram
of
Proposition
”
1.12,
(ii),
is
cartesian,
together
with
the
fact
that
the
“canonical
section”
of
“ζ
=
that
appears
in
Proposition
1.12,
(iii),
is
completely
determined
[cf.
Proposition
1.12,
(v);
Lemma
1.11]
by
the
condition
that
it
lie
under
an
arbitrary
“equivariant
section”
[cf.
Proposition
1.12,
(iv)]
of
the
“ζ
=
”
associated
to
coverings
“X
→
X
”
arising
from
arbitrarily
small
open
subgroups
Π
X
⊆
Π
X
,
it
follows
that
the
isomor-
∼
phism
S
∞
→
T
∞
just
obtained
is
compatible
with
the
pro-fundamental
inclusions
D
X
→
S
∞
,
D
Y
→
T
∞
.
In
particular,
by
Proposition
1.14,
(ii)
[cf.
also
Proposition
1.12,
(i)],
we
conclude
that
α
induces
an
isomorphism
[well-defined
up
to
cuspidally
inner
automorphisms]
∼
c-ab
(S
∞
∼
=
)
Π
c-ab
U
X×X
→
Π
U
Y
×Y
(
∼
=
T
∞
)
c-ab
which
is
compatible
with
the
natural
inclusions
D
X
→
Π
c-ab
U
X×X
,
D
Y
→
Π
U
Y
×Y
.
Finally,
the
functoriality
of
this
isomorphism
follows
from
the
naturality
of
its
construction.
Remark
1.16.1.
It
follows
immediately
from
the
naturality
of
the
constructions
used
in
the
proof
of
Theorem
1.16,
(iii),
that
when
“α”
arises
from
an
isomorphism
∼
of
schemes
X
→
Y
,
the
resulting
α
c-ab
of
Theorem
1.16,
(iii),
coincides
with
the
morphism
induced
on
fundamental
groups
by
the
resulting
isomorphism
of
schemes
∼
U
X×X
→
U
Y
×Y
.
28
SHINICHI
MOCHIZUKI
Lemma
1.17.
(Normal
Subgroups
of
the
Absolute
Galois
Group
of
a
Nonarchimedean
Local
Field)
Let
k
be
a
nonarchimedean
local
field
of
residue
characteristic
p;
write
G
k
for
the
absolute
Galois
group
of
k.
Also,
let
us
write
I
⊆
G
k
for
the
inertia
subgroup
of
G
k
and
W
⊆
I
for
the
wild
inertia
subgroup.
[Here,
we
recall
that
W
is
the
unique
Sylow
pro-p
subgroup
of
I.]
Let
H
⊆
G
k
be
a
closed
subgroup
that
satisfies
[at
least]
one
of
the
following
four
conditions:
(a)
H
is
a
finite
group.
(b)
H
commutes
with
W
.
(c)
H
is
a
pro-prime-to-p
group
[i.e.,
the
order
of
every
finite
quotient
group
of
H
is
prime
to
p]
that
is
normal
in
G
k
.
(d)
H
is
a
topologically
finitely
generated
pro-p
group
that
is
normal
in
G
k
.
Then
H
=
{1}.
Proof.
Indeed,
suppose
that
H
satisfies
condition
(a).
Then
the
fact
that
H
=
{1}
follows
from
[NSW],
Corollary
12.1.3,
Theorem
12.1.7.
Now
suppose
that
H
satisfies
condition
(b).
Then
by
the
well-known
functorial
isomorphism
[arising
from
local
class
field
theory]
between
the
additive
group
underlying
a
finite
field
extension
of
k
that
corresponds
to
an
open
subgroup
J
⊆
G
k
and
the
tensor
product
with
Q
p
of
the
image
of
W
J
in
the
abelianization
J
ab
,
it
follows
immediately
that
the
conjugation
action
of
H
on
W
is
nontrivial,
whenever
H
is
nontrivial.
Thus
we
conclude
again
that
H
=
{1}.
Next,
suppose
that
H
satisfies
condition
(c).
Then
since
H,
W
are
both
normal
in
G
k
,
it
follows
[by
considering
commutators
of
elements
of
H
with
elements
of
W
]
that
arbitrary
elements
of
H
commute
with
arbitrary
elements
of
W
.
In
particular,
H
satisfies
condition
(b),
so
we
conclude
yet
again
that
H
=
{1}.
Finally,
we
assume
that
H
is
nontrivial
and
satisfies
condition
(d).
Then
I
claim
that
H
has
trivial
image
Im(H)
in
G
k
/W
.
Indeed,
since
I/W
,
Im(H)
are
normal
in
G
k
/W
,
and,
moreover,
I/W
is
pro-prime-to-p,
it
follows
that
these
two
groups
commute.
On
the
other
hand,
since,
as
is
well-known,
G
k
/I
acts
faithfully
[by
conjugation,
via
the
cyclotomic
character]
on
I/W
,
it
thus
follows
that
Im(H)
is
trivial,
as
asserted.
Thus,
H
⊆
W
.
Since
[as
in
well-known
—
cf.,
e.g.,
the
proof
of
[Mzk4],
Lemma
15.6]
W
is
a
free
pro-p
group
of
infinite
rank,
we
thus
conclude
that
there
exists
an
open
subgroup
U
⊆
W
[so
U
is
also
a
free
pro-p
group
of
infinite
rank]
containing
H
such
that
the
natural
map
H
ab
⊗
F
p
→
U
ab
⊗
F
p
is
injective,
but
not
surjective.
Then
it
follows
immediately
from
the
well-known
theory
of
free
pro-p
groups
that
there
exists
a
set
of
free
topological
generators
{ξ
i
}
i∈I
[so
the
index
set
I
is
infinite]
of
U
such
that
for
some
finite
subset
J
⊆
I,
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
29
the
elements
{ξ
j
}
j∈J
lie
in
and
topologically
generate
H.
On
the
other
hand,
since
H
is
normal
in
U
,
it
follows
from
the
well-known
structure
of
free
pro-p
groups
that
we
obtain
a
contradiction.
This
completes
the
proof
of
Lemma
1.17.
Remark
1.17.1.
The
author
would
like
to
thank
A.
Tamagawa
for
informing
him
of
the
content
of
Lemma
1.17.
Definition
1.18.
In
the
situation
of
Theorem
1.16,
(i),
(ii),
suppose
further
that
def
Σ
X
=
Σ
Y
;
write
Σ
=
Σ
X
=
Σ
Y
.
(i)
If,
for
every
finite
étale
covering
X
→
X
of
X
arising
from
an
open
subgroup
Π
X
⊆
Π
X
,
it
holds
that
the
map
from
(X
)
cl+
[cf.
§0]
to
conjugacy
classes
of
closed
subgroups
of
Π
X
given
by
assigning
to
a
closed
point
its
associated
decomposition
group
is
injective,
then
we
shall
say
that
X
is
Σ-separated.
(ii)
If
the
map
induced
by
α
on
closed
subgroups
of
Π
X
,
Π
Y
induces
a
bijection
between
the
decomposition
groups
of
the
points
of
X
cl+
,
Y
cl+
,
then
we
shall
say
that
α
is
quasi-point-theoretic.
If
α
is
quasi-point-theoretic,
and,
moreover,
X,
Y
are
Σ-separated
—
in
which
case
α
induces
bijections
∼
X
cl
→
Y
cl
;
∼
X
cl+
→
Y
cl+
—
then
we
shall
say
that
α
is
point-theoretic.
(iii)
Suppose
further
that
we
are
in
the
finite
field
case.
Then
we
shall
say
∼
that
α
is
Frobenius-preserving
if
the
isomorphism
G
k
X
→
G
k
Y
induced
by
α
[cf.
Theorem
1.16,
(ii)]
maps
the
Frobenius
element
of
G
k
X
to
the
Frobenius
element
of
G
k
Y
.
Remark
1.18.1.
In
the
finite
field
case,
when
Σ
†
=
Primes
†
,
the
Frobenius
element
of
G
k
X
may
be
characterized
as
in
[Tama],
Proposition
3.4,
(i),
(ii);
a
similar
statement
holds
for
the
Frobenius
element
of
G
k
Y
.
[Moreover,
in
the
proper
case,
the
Frobenius
element
of
G
k
X
may
be
characterized
as
the
element
of
G
k
X
that
acts
on
M
X
via
multiplication
by
the
cardinality
of
k
X
,
i.e.,
the
cardinality
of
H
1
(G
k
X
,
M
X
)
plus
1.]
Thus,
when
Σ
†
=
Primes
†
,
any
α
as
in
Theorem
1.16,
(i),
(ii),
is
automatically
Frobenius-preserving.
Remark
1.18.2.
Let
us
suppose
that
we
are
in
the
situation
of
Definition
1.18,
and
that
the
base
fields
k
X
,
k
Y
are
finite.
Let
us
refer
to
as
a
quasi-section
[of
Π
X
G
k
X
]
any
closed
subgroup
D
⊆
Π
X
[i.e.,
such
as
a
decomposition
group
of
a
point
∈
X
cl
]
that
maps
isomorphically
onto
an
open
subgroup
of
G
k
X
.
Let
us
refer
to
a
quasi-section
of
Π
X
G
k
X
as
a
subdecomposition
group
if
it
is
contained
in
some
decomposition
group
of
a
point
∈
X
cl
.
(i)
Since
X
is
not
necessarily
Σ-separated,
it
is
not
necessarily
the
case
that
decomposition
groups
of
points
∈
X
cl
are
commensurably
terminal
in
Π
X
[cf.
Propo-
sition
2.6,
(ii),
below].
On
the
other
hand,
if
D
⊆
Π
X
is
a
quasi-section,
and
we
30
SHINICHI
MOCHIZUKI
def
write
E
=
C
Π
X
(D)
⊆
Π
X
for
the
commensurator
of
D
in
Π
X
[cf.
§0],
then
one
ver-
ifies
immediately
E
is
also
a
quasi-section.
[Indeed,
by
considering
the
projection
Π
X
G
k
X
,
it
follows
immediately
that
every
element
of
E
centralizes
some
open
subgroup
D
⊆
D;
on
the
other
hand,
by
considering
the
well-known
properties
of
the
action
of
open
subgroups
of
G
k
on
abelianizations
of
open
subgroups
of
Δ
X
[i.e.,
more
precisely,
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
—
cf.,
e.g.,
[Mumf],
p.
206],
it
follows
that
every
centralizer
of
D
in
Δ
X
is
trivial,
i.e.,
that
E
Δ
X
=
{1}.]
(ii)
It
is
immediate
that
any
maximal
subdecomposition
group
of
Π
X
is,
in
fact,
a
decomposition
group
of
some
point
∈
X
cl
.
On
the
other
hand,
since
X
is
not
necessarily
Σ-separated,
it
is
not
clear
whether
or
not
every
decomposition
group
of
a
point
∈
X
cl
is
necessarily
a
maximal
subdecomposition
group.
If
X,
Y
are
Σ-separated,
then
the
arguments
of
[Tama],
Corollary
2.10,
Proposition
3.8,
yield
a
“group-theoretic”
characterization
of
the
subdecomposition
groups
[hence
also
of
the
maximal
subdecomposition
groups,
i.e.,
the
decomposition
groups
of
points
∈
X
cl
]
of
Π
X
,
Π
Y
in
terms
of
the
actions
of
the
Frobenius
elements.
That
is
to
say,
if
X,
Y
are
Σ-separated,
then
any
Frobenius-preserving
isomorphism
α
is
[quasi-]point-
theoretic.
(iii)
Nevertheless,
as
was
pointed
out
to
the
author
by
A.
Tamagawa,
even
if
X,
Y
are
not
necessarily
Σ-separated,
it
is
still
possible
to
conclude,
essentially
from
the
arguments
of
[Tama],
Corollary
2.10,
Proposition
3.8,
that:
Any
Frobenius-preserving
isomorphism
α
is
quasi-point-theoretic.
Indeed,
it
suffices
to
give
a
“group-theoretic”
characterization
of
the
quasi-sections
D
⊆
Π
X
which
are
decomposition
groups
of
points
∈
X
cl
.
We
may
assume
[for
def
simplicity]
without
loss
of
generality
that
X,
Y
are
proper.
Write
E
=
C
Π
X
(D);
k
D
,
k
E
for
the
finite
extension
fields
of
k
X
determined
by
D,
E.
Let
H
⊆
Δ
X
be
a
characteristic
open
subgroup;
denote
by
Y
→
X
the
covering
determined
by
the
open
subgroup
E
·
H
⊆
Π
X
.
Then
it
follows
immediately
from
the
definition
of
a
“decomposition
group”
that
it
suffices
to
give
a
“group-theoretic”
criterion
for
the
condition
that
Y
(k
D
)
contain
a
point
whose
field
of
definition
[which
is,
a
priori,
some
subextension
in
k
D
of
k
E
]
is
equal
to
k
D
.
In
[Tama],
the
Lefschetz
trace
formula
is
applied
to
compute
the
cardinality
of
Y
(k
D
).
On
the
other
hand,
if
we
use
the
superscript
“fld-def”
to
denote
the
subset
of
points
whose
field
of
definition
is
equal
to
the
field
given
in
parentheses,
and
“|
−
|”
to
denote
the
cardinality
of
a
finite
set,
then
for
any
subextension
k
⊆
k
D
of
k
E
,
we
have
|Y
(k
)|
=
|Y
(k
)
fld-def
|
k
[where
k
⊆
k
ranges
over
the
subextensions
of
k
E
].
In
particular,
by
applying
induction
on
[k
:
k
E
],
one
concludes
immediately
from
the
above
formula
that
|Y
(k
)
fld-def
|
may
be
computed
from
|Y
(k
)|
for
subextensions
k
⊆
k
of
k
E
[while
|Y
(k
)|
may
be
computed,
as
in
[Tama],
from
the
Lefschetz
trace
formula].
This
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
31
yields
the
desired
“group-theoretic”
characterization
of
the
decomposition
groups
of
Π
X
.
Remark
1.18.3.
Note
that
in
the
finite
field
case,
if
α
as
in
Theorem
1.16,
(i),
(ii),
is
Frobenius-preserving,
then
the
cardinalities,
hence
also
the
characteristics,
of
k
X
,
k
Y
coincide.
Indeed,
this
follows
immediately
by
reducing
to
the
proper
case
via
Theorem
1.16,
(i),
and
considering
the
actions
of
G
k
X
,
G
k
Y
[cf.
Theorem
1.16,
∼
(ii)]
on
M
X
,
M
Y
[which
are
compatible
relative
to
the
isomorphism
M
X
→
M
Y
induced
by
α].
Now
we
return
to
the
notation
of
the
discussion
preceding
Theorem
1.16.
Ob-
serve
that
the
automorphism
τ
:
X
×
X
→
X
×
X
given
by
switching
the
two
factors
induces
an
outer
automorphism
of
Π
U
X×X
.
More-
over,
by
choosing
the
basepoints
used
to
form
the
various
fundamental
groups
in-
volved
in
an
appropriate
fashion,
it
follows
that
there
exists
an
automorphism
Π
τ
:
Π
U
X×X
→
Π
U
X×X
among
those
automorphisms
induced
by
τ
[i.e.,
all
of
which
are
related
to
one
another
by
composition
with
an
inner
automorphism]
which
induces
the
automor-
phism
on
Π
X×X
=
Π
X
×
G
k
Π
X
given
by
switching
the
two
factors;
preserves
the
subgroup
D
X
⊆
Π
U
X×X
;
and
preserves
and
induces
the
identity
automorphism
on
the
subgroup
I
X
⊆
D
X
(⊆
Π
U
X×X
).
Note
that
by
the
slimness
of
Proposition
1.8,
(i),
together
with
the
well-known
commensurable
terminality
of
D
X
⊆
Π
U
X×X
in
Π
U
X×X
[cf.,
e.g.,
[the
proof
of]
[Mzk5],
Lemma
1.3.12],
it
follows
that,
at
least
when
Σ
=
Primes,
these
three
conditions
[are
more
than
sufficient
to]
uniquely
determine
Π
τ
,
up
to
composition
with
an
inner
automorphism
arising
from
I
X
;
one
then
obtains
a
natural
Π
τ
for
arbitrary
Σ
[well-defined
up
to
composition
with
an
inner
automorphism
arising
from
I
X
]
by
taking
the
automorphism
induced
on
the
appropriate
quotients
by
“Π
τ
in
the
case
Σ
=
Primes”.
Proposition
1.19.
(Switching
the
Two
Factors)
The
automorphism
c-ab
:
Π
c-ab
Π
c-ab
τ
U
X×X
→
Π
U
X×X
induced
by
Π
τ
is
the
unique
automorphism
of
the
profinite
group
Π
c-ab
U
X×X
,
up
to
composition
with
a
cuspidally
inner
automorphism,
that
satisfies
the
following
two
conditions:
(a)
it
preserves
the
quotient
Π
c-ab
U
X×X
Π
X×X
and
induces
on
this
quotient
the
automorphism
on
Π
X×X
=
Π
X
×
G
k
Π
X
given
by
switching
the
two
factors;
(b)
it
preserves
the
image
of
I
X
⊆
D
X
→
Π
c-ab
U
X×X
.
Proof.
This
follows
immediately
from
Proposition
1.15,
(i).
32
SHINICHI
MOCHIZUKI
Section
2:
Points
and
Functions
We
maintain
the
notation
of
§1
[i.e.,
the
discussion
preceding
Theorem
1.16].
If
x
∈
X
cl
,
then
we
shall
denote
by
D
x
⊆
Π
X
the
decomposition
group
of
x
[well-defined
up
to
conjugation
in
Π
X
].
If
x
∈
X(k),
then
D
x
determines
a
section
s
x
:
G
k
→
Π
X
[which
is
well-defined
as
a
geometrically
outer
homomorphism].
Next,
let
S
⊆
X
cl
be
a
finite
set.
If
n
is
a
Σ
†
-integer
[cf.
§0],
then
the
Kummer
exact
sequence
1
→
μ
n
→
G
m
→
G
m
→
1
[where
G
m
→
G
m
is
the
n-th
power
map;
μ
n
is
defined
so
as
to
make
the
sequence
exact]
on
the
étale
site
of
X
determines
a
homomorphism
Pic(X)
→
H
2
(Δ
X
,
μ
n
)
[where
Pic(X)
is
the
Picard
group
of
X].
Now
there
is
a
unique
isomorphism
∼
μ
n
→
M
X
/n
·
M
X
such
that
the
homomorphism
Pic(X)
→
H
2
(Δ
X
,
μ
n
)
sends
line
bundles
of
degree
1
to
the
element
determined
by
1
∈
Z/nZ
via
the
composite
of
the
induced
iso-
∼
morphism
H
2
(Δ
X
,
μ
n
)
→
H
2
(Δ
X
,
M
X
/n
·
M
X
)
with
the
tautological
isomorphism
∼
H
2
(Δ
X
,
M
X
/n·M
X
)
→
Z/nZ
[cf.
Proposition
1.2,
(i)].
In
the
following
discussion,
we
shall
identify
μ
n
with
M
X
/n
·
M
X
via
this
isomorphism.
If
we
consider
the
Kummer
exact
sequence
on
the
étale
site
of
U
S
⊆
X
[and
pass
to
the
inverse
limit
with
respect
to
n],
then
we
obtain
a
natural
homomorphism
×
Γ(U
S
,
O
U
)
→
H
1
(Π
U
S
,
M
X
)
S
[where
we
note
that
here,
it
suffices
to
consider
the
group
cohomology
of
Π
U
S
[i.e.,
as
opposed
to
the
étale
cohomology
of
U
S
],
since
the
extraction
of
n-th
roots
of
an
×
)
yields
finite
étale
coverings
of
U
S
that
correspond
to
open
element
of
Γ(U
S
,
O
U
S
×
subgroups
of
Π
U
S
]
which
is
injective
[since
the
abelian
topological
group
Γ(U
S
,
O
U
)
S
†
is
clearly
topologically
finitely
generated
and
free
of
p
-torsion,
hence
injects
into
its
prime-to-p
†
completion]
whenever
Σ
†
=
Primes
†
.
In
particular,
by
allowing
S
to
vary,
we
obtain
a
natural
homomorphism
×
1
K
X
→
lim
−→
H
(Π
U
S
,
M
X
)
S
[where
K
X
is
the
function
field
of
X;
the
direct
limit
is
over
all
finite
subsets
S
of
X
cl
]
which
is
injective
whenever
Σ
†
=
Primes
†
.
Proposition
2.1.
subset,
write
(Kummer
Classes
of
Functions)
If
S
⊆
X
cl
is
a
finite
c-cn
Δ
U
S
Δ
c-ab
U
S
Δ
U
S
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
33
for
the
maximal
cuspidally
abelian
and
maximal
cuspidally
central
quo-
tients,
respectively,
and
c-cn
Π
U
S
Π
c-ab
U
S
Π
U
S
for
the
corresponding
quotients
of
Π
U
S
.
If
x
∈
X
cl
,
then
let
us
write
D
x
[U
S
]
⊆
Π
U
S
for
the
decomposition
group
of
x
in
Π
U
S
[which
is
well-defined
up
to
conjugation
in
Π
U
S
]
and
I
x
[U
S
]
⊆
D
x
[U
S
]
for
the
inertia
subgroup.
[Thus,
when
x
∈
S,
we
def
obtain
[cf.
Proposition
1.6,
(ii),
(iii)]
a
natural
isomorphism
of
M
X
with
I
x
[U
S
]
=
D
x
[U
S
]
Δ
U
S
.]
(i)
The
natural
surjections
induce
isomorphisms
as
follows:
∼
∼
1
c-ab
1
H
1
(Π
c-cn
U
S
,
M
X
)
→
H
(Π
U
S
,
M
X
)
→
H
(Π
U
S
,
M
X
)
In
particular,
we
obtain
natural
homomorphisms
as
follows:
∼
∼
×
1
c-ab
1
)
→
H
1
(Π
c-cn
Γ(U
S
,
O
U
U
S
,
M
X
)
→
H
(Π
U
S
,
M
X
)
→
H
(Π
U
S
,
M
X
)
S
∼
∼
×
1
c-cn
1
c-ab
1
→
lim
K
X
−→
H
(Π
U
S
,
M
X
)
→
lim
−→
H
(Π
U
S
,
M
X
)
→
lim
−→
H
(Π
U
S
,
M
X
)
S
S
S
These
natural
homomorphisms
are
injective
whenever
Σ
†
=
Primes
†
.
(ii)
Suppose
that
S
⊆
X(k)
is
a
finite
subset.
Then
restricting
cohomology
classes
of
Π
U
S
to
the
various
I
x
[U
S
],
for
x
∈
S,
yields
a
natural
exact
sequence
∧
1
→
(k
×
)
→
H
1
(Π
U
S
,
M
X
)
→
†
Z
x∈S
†
].
Moreover,
the
image
[via
the
[where
we
identify
Hom
Z
†
(I
x
[U
S
],
M
X
)
with
Z
∧
×
natural
homomorphism
given
in
(i)]
of
Γ(U
S
,
O
U
)
in
H
1
(Π
U
S
,
M
X
)/(k
×
)
is
equal
S
∧
to
the
inverse
image
in
H
1
(Π
U
S
,
M
X
)/(k
×
)
of
the
submodule
of
x∈S
†
Z
⊆
Z
x∈S
determined
by
the
principal
divisors
[with
support
in
S].
A
similar
statement
c-cn
holds
when
“Π
U
S
”
is
replaced
by
“Π
c-ab
U
S
”
or
“Π
U
S
”.
×
(iii)
If
f
∈
Γ(U
S
,
O
U
),
write
S
∈
H
1
(Π
c-cn
κ
c-cn
f
U
S
,
M
X
);
κ
c-ab
∈
H
1
(Π
c-ab
f
U
S
,
M
X
);
κ
f
∈
H
1
(Π
U
S
,
M
X
)
for
the
associated
Kummer
classes.
If
x
∈
X
cl
\S,
then
D
x
[U
S
]
maps,
via
the
natural
surjection
Π
U
S
G
k
,
isomorphically
onto
the
open
subgroup
G
k(x)
⊆
G
k
34
SHINICHI
MOCHIZUKI
[where
k(x)
is
the
residue
field
of
X
at
x].
Moreover,
the
images
of
the
pulled
back
classes
∼
|
D
x
[U
S
]
=
κ
c-ab
|
D
x
[U
S
]
=
κ
f
|
D
x
[U
S
]
∈
H
1
(D
x
[U
S
],
M
X
)
→
H
1
(G
k(x)
,
M
X
)
κ
c-cn
f
f
∼
∧
→
(k(x)
×
)
∧
∧
in
(k(x)
×
)
are
equal
to
the
image
in
(k(x)
×
)
of
the
value
of
f
at
x.
Proof.
Assertion
(i)
follows
immediately
from
the
definitions.
The
exact
sequence
of
assertion
(ii)
follows
immediately
from
Proposition
1.4,
(ii).
The
characterization
×
)
is
immediate
from
the
definitions
and
the
exact
sequence
of
the
image
of
Γ(U
S
,
O
U
S
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
the
definitions
and
the
functoriality
of
the
Kummer
class.
Remark
2.1.1.
If,
in
the
situation
of
Proposition
2.1,
(iii),
we
think
of
the
extension
of
Π
c-cn
of
Π
X
as
being
given
by
the
extension
D
S
[cf.
Proposition
1.8,
U
S
(iii)],
where
D
is
a
fundamental
extension
of
Π
X×X
that
appears
as
a
quotient
of
∧
Π
U
X×X
[hence
is
“rigid”
with
respect
to
the
action
of
(k
×
)
—
cf.
Proposition
1.9,
(iii);
the
proof
of
Theorem
1.16,
(iii)],
then
it
follows
that
the
image
of
D
x
[U
S
]
in
Π
c-cn
U
S
may
be
thought
of
as
the
image
of
D
x
[U
S
]
in
D
S
.
If,
moreover,
we
assume,
for
simplicity,
that
x
∈
X(k),
S
⊆
X(k),
then
this
image
of
D
x
[U
S
]
in
D
S
amounts
to
a
section
of
D
S
Π
X
G
k
lying
over
the
section
s
x
of
Π
X
G
k
.
Since
D
S
is
defined
as
a
certain
fiber
product,
this
section
is
equivalent
to
a
collection
of
sections
[regarded
as
cyclotomically
outer
homomorphisms]
γ
y,x
:
G
k
→
D
y,x
[where
y
ranges
over
the
points
of
S].
[Here,
we
note
that
it
is
immediate
from
the
definitions
that,
as
the
notation
suggests,
γ
y,x
depends
only
on
x,
y
—
i.e.,
that
γ
y,x
is
independent
of
the
choice
of
S.]
That
is
to
say,
from
this
point
of
view,
Proposition
2.1,
(iii),
may
be
regarded
as
stating
that:
∧
∧
×
)
The
image
in
(k
×
)
=
(k(x)
×
)
of
the
value
of
a
function
∈
Γ(U
S
,
O
U
S
at
x
∈
X(k)
may
be
computed
from
its
Kummer
class,
as
soon
as
one
knows
the
sections
γ
y,x
:
G
k
→
D
y,x
,
for
y
∈
S.
Also,
before
proceeding,
we
note
that
an
arbitrary
section
of
D
y,x
G
k
differs
[as
a
cyclotomically
outer
homomorphism]
from
γ
y,x
by
the
action
of
an
element
of
∼
∧
H
1
(G
k
,
M
X
)
→
(k
×
)
.
Thus,
the
datum
of
“γ
y,x
”
may
be
regarded
as
a
trivializa-
∧
tion
of
a
certain
(k
×
)
-torsor.
Remark
2.1.2.
The
finite
field
portion
of
Proposition
2.1
may
be
regarded
as
the
evident
finite
field
analogue
of
[a
certain
portion
of]
the
theory
of
[Mzk8],
§4.
Also,
we
observe
that
the
approach
of
“reconstructing
the
function
field
of
the
curve
via
Kummer
theory,
as
opposed
to
class
field
theory
[as
was
done
in
[Tama],
[Uchi]]”
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
35
has
the
advantage
of
being
applicable
to
nonarchimedean
local
fields,
as
well
as
to
finite
fields.
Definition
2.2.
For
x,
y
∈
X(k),
we
shall
refer
to
the
section
[regarded
as
a
cyclotomically
outer
homomorphism]
γ
y,x
:
G
k
→
D
y,x
as
the
Green’s
trivialization
of
D
at
(y,
x).
If
D
is
a
divisor
on
X
supported
in
the
subset
of
k-rational
points
X(k)
⊆
X
cl
,
then
multiplication
of
the
various
Green’s
trivializations
for
the
points
in
the
support
of
D
determines
a
section
[regarded
as
a
cyclotomically
outer
homomorphism]
γ
D,x
:
G
k
→
D
D,x
which
we
shall
refer
to
as
the
Green’s
trivialization
of
D
at
(D,
x).
[Note
that
the
definition
of
γ
D,x
generalizes
immediately
to
the
case
where
the
divisor
D,
but
not
necessarily
the
points
in
its
support,
is
rational
over
k
—
cf.
Remark
1.10.1.]
Remark
2.2.1.
The
terminology
of
Definition
2.2,
is
intended
to
suggest
the
similarity
between
the
γ
y,x
of
the
present
discussion
and
the
“Green’s
functions”
that
occur
in
the
theory
of
bipermissible
metrics
—
cf.,
e.g.,
[MB],
§4.11.4.
Remark
2.2.2.
Note
that
the
Green’s
trivializations
are
symmetric
with
respect
of
Proposition
1.19.
to
the
involution
of
D
induced
by
the
automorphism
Π
c-ab
τ
Indeed,
relative
to
the
natural
projections
Π
U
X×X
Π
c-ab
U
X×X
D
the
Green’s
trivialization
at
(y,
x)
is
simply
the
section
of
D
G
k
arising
[by
composition]
from
the
section
of
Π
U
X×X
G
k
determined
by
the
decomposition
group
of
the
point
(y,
x)
∈
U
X×X
(k).
Thus,
the
asserted
symmetry
of
the
Green’s
is
compatible
with
Π
τ
,
together
with
trivializations
follows
from
the
fact
that
Π
c-ab
τ
the
evident
fact
that
[by
“transport
of
structure”]
Π
τ
maps
the
decomposition
group
of
(y,
x)
∈
U
X×X
(k)
isomorphically
onto
the
decomposition
group
of
(x,
y)
∈
U
X×X
(k).
If
d
∈
Z,
denote
by
J
d
the
subscheme
of
the
Picard
scheme
of
X
that
parame-
def
trizes
line
bundles
of
degree
d;
write
J
=
J
0
.
Thus,
J
d
is
a
torsor
over
J.
Note
that
there
is
a
natural
morphism
X
→
J
1
[given
by
assigning
to
a
point
of
X
the
line
bundle
of
degree
1
determined
by
the
point].
Thus,
the
basepoint
of
X
[already
chosen
in
§1]
determines
a
basepoint
of
J
1
.
At
the
level
of
“geometrically
pro-Σ”
étale
fundamental
groups,
this
morphism
induces
a
surjective
homomorphism
Π
X
Π
J
1
36
SHINICHI
MOCHIZUKI
whose
kernel
is
the
kernel
of
the
maximal
abelian
quotient
Δ
X
Δ
ab
X
.
In
partic-
ular,
for
x
∈
X(k),
the
section
s
x
determines
a
section
t
x
:
G
k
→
Π
J
1
.
Note
that
applying
the
“change
of
structure
group”
given
by
the
“multiplication
by
d
map”
on
J
to
the
J-torsor
J
1
yields
the
J-torsor
J
d
.
[Indeed,
this
follows
by
considering
the
group
structure
of
the
Picard
scheme.]
Thus,
we
obtain
a
morphism
J
1
→
J
d
whose
induced
morphism
on
fundamental
groups
Π
J
1
→
Π
J
d
determines
an
isomorphism
of
Π
J
d
with
the
push-forward
of
the
extension
Π
J
1
[i.e.,
ab
ab
of
G
k
by
Δ
J
1
∼
=
Δ
ab
X
]
via
the
homomorphism
Δ
X
→
Δ
X
given
by
multiplication
by
d.
When
d
≥
1,
the
group
structure
on
the
Picard
scheme
also
determines
a
morphism
Π
J
1
→
Π
J
d
[where
the
product
is
a
fiber
product
over
G
k
of
d
factors
of
Π
J
1
]
which
determines
an
isomorphism
of
Π
J
d
with
the
push-forward
ab
of
the
ab
extension
constituted
by
the
fiber
product
via
the
homomorphism
Δ
X
→
Δ
X
[i.e.,
from
a
product
of
d
ab
ab
copies
of
Δ
X
to
Δ
X
given
by
adding
up
the
d
components].
Moreover,
one
verifies
immediately
that
when
d
≥
1,
these
two
constructions
of
“Π
J
d
”
from
Π
J
1
yield
groups
that
are
naturally
isomorphic.
Thus,
by
applying
the
various
homomorphisms
induced
on
fundamental
groups
by
the
group
structure
of
the
Picard
scheme,
it
follows
that
if
D
is
any
divisor
of
degree
d
on
X
whose
support
lies
in
the
set
of
k-rational
points
X(k)
⊆
X
cl
,
then
D
determines
a
section
t
D
:
G
k
→
Π
J
d
which
may
be
constructed
entirely
group-theoretically
from
the
“t
x
”,
where
x
∈
X(k)
ranges
over
the
points
in
the
support
of
D.
In
particular,
if
D
is
of
degree
0,
then
the
section
t
D
:
G
k
→
Π
J
may
be
compared
with
the
identity
section
of
Π
J
to
obtain
a
cohomology
class:
η
D
∈
H
1
(G
k
,
Δ
ab
X
)
Now
we
have
the
following
well-known
result:
Proposition
2.3.
(Points
and
Galois
Sections)
Suppose
that
Σ
=
Primes.
Then,
in
the
notation
of
the
above
discussion:
(i)
The
divisor
D
is
principal
if
and
only
if
η
D
=
0.
(ii)
The
map
x
→
D
x
from
X
cl
to
conjugacy
classes
of
closed
subgroups
of
Π
X
is
injective,
i.e.,
X
is
Primes-separated.
Proof.
First,
we
consider
assertion
(i).
By
well-known
general
nonsense
[cf.,
e.g.,
[Naka],
Claim
(2.2);
[NTs],
Lemma
(4.14);
[Mzk4],
the
Remark
preceding
Definition
6.2],
there
is
a
natural
isomorphism
∼
∧
H
1
(k,
Δ
ab
X
)
→
J(k)
(⊇
J(k))
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
37
[where
the
“∧”
denotes
the
profinite
completion]
which
maps
η
D
to
the
element
of
J(k)
determined
by
D.
[Here,
we
recall
that
this
natural
isomorphism
arises
by
considering
the
long
exact
sequence
obtained
by
applying
the
functors
H
∗
(G
k
,
−)
to
the
short
exact
sequence
of
G
k
-modules
1
→
J(k)[n]
→
J(k)
→
J(k)
→
1
—
where
n
is
a
positive
integer;
the
morphism
J(k)
→
J(k)
is
the
“multiplication
by
n
map”;
J(k)[n]
is
defined
so
as
to
make
the
sequence
exact.]
Thus,
assertion
(i)
follows
immediately.
To
prove
assertion
(ii),
it
suffices
[by
possibly
base-changing
to
a
finite
exten-
sion
of
k]
to
verify
that
two
points
x
1
,
x
2
∈
X(k)
that
induce
Δ
X
-conjugate
sections
s
x
1
,
s
x
2
are
necessarily
equal
[cf.
also
[Tama],
Corollary
2.10].
But
this
follows
for-
mally
from
assertion
(i),
by
considering
the
divisor
x
1
−
x
2
[and
the
well-known
fact
that
the
natural
morphism
X
→
J
1
considered
above
is
an
embedding].
Remark
2.3.1.
From
the
point
of
view
of
Definition
1.7,
(ii),
the
reader
may
feel
tempted
to
expect
that
[still
under
the
assumption
that
Σ
=
Primes]
D
is
principal
if
and
only
if
the
extension
D
D
of
Π
X
[by
M
X
]
is
trivial
[i.e.,
determines
the
zero
class
in
H
2
(Π
X
,
M
X
)].
When
k
is
nonarchimedean
local,
it
is
not
difficult
to
verify,
using
Proposition
2.3,
(i),
that
this
is
indeed
the
case.
On
the
other
hand,
when
k
is
finite,
although
this
condition
for
principality
is
easily
verified
to
be
necessary,
it
is
not,
however,
sufficient,
since
it
only
involves
the
“prime-to-p
†
portion”
of
the
point
of
J(k)
determined
by
D.
Definition
2.4.
In
the
situation
of
Theorem
1.16,
(iii),
suppose
further
that
def
(Σ
=
)
Σ
X
=
Σ
Y
,
and
that
α
is
point-theoretic.
Let
S
⊆
X
cl
be
a
[not
necessarily
∼
finite]
subset
that
corresponds
via
the
bijection
X
cl
→
Y
cl
induced
by
[the
point-
theoreticity
of]
α
to
a
subset
T
⊆
Y
cl
.
(i)
Write
D
(respectively,
E)
for
the
fundamental
extension
of
Π
X×X
(respec-
c-ab
tively,
Π
Y
×Y
)
that
arises
as
the
quotient
of
Π
c-ab
U
X×X
(respectively,
Π
U
Y
×Y
)
by
the
c-cn
kernel
of
the
maximal
cuspidally
central
quotient
Δ
c-ab
U
X×X
Δ
U
X×X
(respectively,
c-cn
c-ab
Δ
c-ab
induces
an
isomorphism:
U
Y
×Y
Δ
U
Y
×Y
)
[cf.
Proposition
1.8,
(iv)].
Thus,
α
∼
α
c-cn
:
D
→
E
We
shall
say
that
α
is
(S,
T
)-locally
Green-compatible
if,
for
every
pair
of
points
(x
1
,
x
2
)
∈
X(k
X
)
×
X(k
X
)
corresponding
via
the
bijection
induced
by
α
to
a
pair
of
points
(y
1
,
y
2
)
∈
Y
(k
Y
)
×
Y
(k
Y
),
such
that
x
2
∈
S,
y
2
∈
T
,
the
isomorphism
∼
D
x
1
,x
2
→
E
y
1
,y
2
[obtained
by
restricting
α
c-cn
]
is
compatible
with
the
Green’s
trivializations.
We
shall
say
that
α
is
(S,
T
)-locally
degree
zero
(respectively,
(S,
T
)-locally
principally)
38
SHINICHI
MOCHIZUKI
Green-compatible
if,
for
every
x
∈
X(k
X
)
S
and
every
divisor
of
degree
zero
(respectively,
principal
divisor)
D
supported
in
X(k
X
)
⊆
X
cl
corresponding
via
the
bijection
induced
by
α
to
a
pair
(y,
E)
of
Y
[so
y
∈
Y
(k
Y
)
T
],
the
isomorphism
∼
D
D,x
→
E
E,y
is
compatible
with
the
Green’s
trivializations.
(ii)
We
shall
say
that
α
is
totally
(S,
T
)-locally
Green-compatible
(respectively,
totally
(S,
T
)-locally
degree
zero
Green-compatible;
totally
(S,
T
)-locally
principally
Green-compatible)
if,
for
all
pairs
of
connected
finite
étale
coverings
X
→
X,
Y
→
Y
that
arise
from
open
subgroups
of
Π
X
,
Π
Y
that
correspond
via
α,
the
isomorphism
∼
Π
X
→
Π
Y
induced
by
α
is
(S
,
T
)-locally
Green-compatible
(respectively,
(S
,
T
)-locally
de-
gree
zero
Green-compatible;
(S
,
T
)-locally
principally
Green-compatible),
where
S
⊆
(X
)
cl
,
T
⊆
(Y
)
cl
are
the
inverse
images
in
X
,
Y
of
S,
T
,
respectively.
(iii)
With
respect
to
the
terminology
introduced
in
(i),
(ii),
when
S
=
X
cl
,
T
=
Y
cl
,
then
we
shall
replace
the
phrase
“(S,
T
)-locally”
by
the
phrase
“globally”.
Remark
2.4.1.
In
the
situation
of
Definition
2.4,
if
X
→
X,
Y
→
Y
are
con-
nected
finite
étale
coverings
that
arise
from
open
subgroups
of
Π
X
,
Π
Y
that
corre-
∼
spond
via
α;
D
→
E
is
the
isomorphism
of
fundamental
extensions
of
Π
X×X
,
Π
Y
×Y
that
arises
from
the
isomorphism
α
c-ab
of
Theorem
1.16,
(iii);
and
the
points
x
1
,
x
2
(respectively,
y
1
,
y
2
)
are
Δ
X
-
(respectively,
Δ
Y
-)
conjugate,
then
it
follows
imme-
diately
from
the
compatibility
of
α
c-ab
with
the
natural
inclusions
D
X
→
Π
c-ab
U
X×X
,
∼
D
Y
→
Π
c-ab
U
Y
×Y
[cf.
Theorem
1.16,
(iii)]
that
the
isomorphism
D
x
1
,x
2
→
E
y
1
,y
2
is
automatically
compatible
with
the
Green’s
trivializations.
[Indeed,
this
follows
from
the
easily
verified
fact
that
the
Green’s
trivializations
in
this
case
are,
in
essence,
”
of
Proposition
1.12.]
specializations
of
conjugates
of
the
“canonical
sections
of
ζ
=
Unfortunately,
however,
the
author
is
unable,
at
the
time
of
writing,
to
see
how
to
generalize
the
argument
applied
in
the
proof
of
Theorem
1.16,
(iii),
involving
Lemma
1.11;
Proposition
1.12,
(v),
so
as
to
cover
the
case
where
the
points
x
1
,
x
2
(respectively,
y
1
,
y
2
)
fail
to
be
Δ
X
-
(respectively,
Δ
Y
-)
conjugate.
Remark
2.4.2.
It
is
immediate
that
(S,
T
)-local
Green-compatibility
(respec-
tively,
(S,
T
)-local
degree
zero
Green-compatibility)
implies
(S,
T
)-local
degree
zero
Green-compatibility
(respectively,
(S,
T
)-local
principal
Green-compatibility),
and
that
total
(S,
T
)-local
Green-compatibility
(respectively,
total
(S,
T
)-local
degree
zero
Green-compatibility)
implies
total
(S,
T
)-local
degree
zero
Green-compatibility
(respectively,
total
(S,
T
)-local
principal
Green-compatibility).
Theorem
2.5.
(Reconstruction
of
Functions)
In
the
situation
of
Theorem
def
1.16,
(iii),
suppose
further
that
(Σ
=
)
Σ
X
=
Σ
Y
,
and
that
α
is
point-theoretic.
Then:
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
39
(i)
Let
S
⊆
X
cl
,
T
⊆
Y
cl
be
finite
subsets
that
correspond
via
the
bijection
∼
X
cl
→
Y
cl
induced
by
α.
Then
α,
α
c-ab
induce
isomorphisms
[well-defined
up
to
cuspidally
inner
automorphisms]
∼
c-ab
Π
c-ab
U
S
→
Π
V
T
def
[where
V
T
=
Y
\T
]
lying
over
α,
which
are
functorial
with
respect
to
α
and
S,
T
,
as
well
as
with
respect
to
passing
to
connected
finite
étale
coverings
of
X,
Y
[that
do
not
necesarily
arise
from
open
subgroups
of
Π
X
,
Π
Y
!].
∼
(ii)
Suppose
that
Σ
=
Primes.
Then
the
bijection
X
cl
→
Y
cl
induced
by
α
in-
duces
a
bijection
between
the
groups
of
principal
divisors
on
X,
Y
.
This
bijection,
together
with
the
isomorphisms
of
(i),
induces
a
compatible
isomorphism
∧
∼
×
×
·
(k
X
)
→
K
Y
×
·
(k
Y
×
)
K
X
∧
between
the
push-forwards
of
the
multiplicative
groups
associated
to
the
function
∧
×
×
∧
fields
of
X,
Y
,
relative
to
the
homomorphisms
k
X
→
(k
X
)
,
k
Y
×
→
(k
Y
×
)
.
Proof.
Assertion
(i)
follows
immediately
by
“specializing
to
S,
T
”
the
isomorphism
of
Theorem
1.16,
(iii)
[cf.
also
Proposition
1.14,
(i),
(ii);
the
definitions
of
the
various
objects
involved].
[Here,
we
note
that
the
functoriality
asserted
in
assertion
(i),
which
is
somewhat
stronger
than
the
functoriality
asserted
in
Theorem
1.16,
(iii),
follows
from
the
definitions,
together
with
the
naturality
of
the
constructions
applied
in
the
proof
of
Theorem
1.16,
(iii)
—
cf.,
e.g.,
the
diagram
of
Proposition
1.9,
(ii).]
Assertion
(ii)
follows
immediately
from
assertion
(i);
Proposition
2.3,
(i);
Proposition
2.1,
(i),
(ii).
∼
c-ab
of
Theorem
Remark
2.5.1.
In
fact,
the
crucial
isomorphism
Π
c-ab
U
S
→
Π
V
T
2.5,
(i),
may
also
be
constructed,
in
the
finite
field
case,
via
the
techniques
to
be
introduced
in
§3
[although
we
shall
not
discuss
this
approach
in
detail;
cf.,
however,
the
proof
of
Theorem
3.10].
On
the
other
hand,
observe
that
unlike
the
techniques
of
§3,
the
techniques
of
§1
[in
particular,
the
proof
of
Theorem
1.16,
(iii),
via
Propositions
1.9,
1.12]
apply
to
situations
[e.g.,
the
case
of
nonarchimedean
local
fields!]
where
the
weight
filtration
[cf.
§3]
does
not
admit
a
Galois-invariant
splitting.
Indeed,
the
techniques
of
§1,
essentially
only
require
that
the
Galois
cohomology
of
the
base
field
admit
a
natural
duality
pairing.
Moreover,
even
in
the
∼
c-ab
finite
field
case,
in
light
of
the
importance
of
this
isomorphism
Π
c-ab
U
S
→
Π
V
T
in
the
theory
of
the
present
paper,
it
is
of
interest
to
see
that
this
isomorphism
may
be
constructed
via
two
fundamentally
different
approaches.
Finally,
although
the
techniques
of
§3
are
better
suited
to
the
reconstruction
of
the
Green’s
trivializations,
they
have
the
drawback
that
they
depend
essentially
on
the
choice
of
a
“basepoint”
x
∗
∈
X(k).
Thus,
it
is
of
interest
to
know
that
this
isomorphism
may
be
constructed
[i.e.,
via
the
techniques
of
§1]
“cohomologically”
[cf.
Proposition
1.6,
(i)]
without
making
such
a
choice.
40
SHINICHI
MOCHIZUKI
Remark
2.5.2.
In
the
case
of
nonarchimedean
local
fields,
it
is
natural
to
ask,
in
the
style
of
[Mzk8],
§4,
whether
or
not
various
“canonical
integral
structures”
on
the
extensions
D
x,y
[where
x,
y
∈
X(k)]
of
G
k
by
M
X
are
preserved
by
arbitrary
isomorphisms
of
arithmetic
fundamental
groups.
When
x
=
y,
such
a
canonical
integral
structure
is
determined
by
the
Green’s
trivialization;
when
x
=
y,
such
a
canonical
integral
structure
is
determined
by
the
integral
structure
[in
the
usual
sense
of
scheme
theory]
on
the
canonical
sheaf
of
the
stable
model
of
the
curve
[when
the
curve
has
stable
reduction]
—
cf.
[Mzk8],
§4.
Before
proceeding,
we
note
the
following
“analogue
for
Π
c-ab
U
S
”
of
Proposition
1.15,
(i):
Proposition
2.6.
(Automorphisms
and
Commensurators)
Let
Π
c-ab
U
S
be
c-ab
as
in
Proposition
2.1.
For
x
∈
S,
write
D
x
[U
S
]
→
Π
U
S
for
the
natural
inclusion.
Then:
(i)
Any
automorphism
α
of
the
profinite
group
Π
c-ab
U
S
which
(a)
is
compatible
with
the
natural
surjection
Π
c-ab
U
S
Π
X
and
induces
the
identity
on
Π
X
;
(b)
for
each
x
∈
S,
preserves
the
image
of
M
X
∼
=
I
x
[U
S
]
⊆
D
x
[U
S
]
via
the
c-ab
natural
inclusion
D
x
[U
S
]
→
Π
U
S
is
cuspidally
inner.
(ii)
Suppose
that
X
is
Σ-separated.
Then
for
x
∈
S,
D
x
is
commensurably
terminal
in
Π
X
.
(iii)
Suppose
that
X
is
Σ-separated.
Then
the
image
of
D
x
[U
S
]
→
Π
c-ab
U
S
is
c-ab
commensurably
terminal
in
Π
U
S
.
Proof.
First,
we
observe
that
assertion
(ii)
follows
formally
from
the
definition
of
a
“decomposition
group”
and
“Σ-separated”.
Thus,
assertion
(i)
(respectively,
(iii))
follows
by
an
argument
which
is
entirely
similar
to
the
argument
that
was
used
to
prove
assertion
(i)
(respectively,
(iii))
of
Proposition
1.15.
Remark
2.6.1.
In
the
situation
of
Definition
2.4,
suppose
that
S,
T
are
finite,
and
that
α
arises
from
an
isomorphism
∼
Π
U
S
→
Π
V
T
which
is
point-theoretic
[or,
equivalently,
quasi-point-theoretic]
—
a
condition
that
is
automatically
satisfied
in
the
finite
field
case
whenever
α
is
Frobenius-preserving
[cf.
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
41
Remark
1.18.2].
Then
observe
that,
[in
light
of
our
point-theoreticity
assumption]
it
follows
from
Proposition
2.6,
(i),
that
the
resulting
induced
isomorphism
∼
c-ab
Π
c-ab
U
S
→
Π
V
T
coincides
[up
to
cuspidally
inner
automorphisms]
with
the
isomorphism
of
Theorem
2.5,
(i).
Thus,
in
light
of
Remark
2.2.2,
it
follows
formally
from
the
definitions
that
α
is
totally
(S,
T
)-locally
Green-compatible.
Corollary
2.7.
(Point-theoretic
Totally
Locally
Principally
Green-
compatible
Isomorphisms)
In
the
situation
of
Theorem
1.16,
(iii),
assume
fur-
def
ther
that
(Σ
=
)
Σ
X
=
Σ
Y
=
Primes,
and
that
α
is
point-theoretic
and
to-
tally
(S,
T
)-locally
principally
Green-compatible,
for
some
nonempty
sub-
∼
sets
S
⊆
X
cl
,
T
⊆
Y
cl
which
correspond
via
the
bijection
X
cl
→
Y
cl
induced
by
α.
Then
α
arises
from
a
uniquely
determined
commutative
diagram
of
schemes
∼
X
⏐
⏐
→
X
→
∼
Y
⏐
⏐
Y
in
which
the
horizontal
arrows
are
isomorphisms;
the
vertical
arrows
are
the
pro-
finite
étale
coverings
determined
by
the
profinite
groups
Π
X
,
Π
Y
.
Proof.
Corollary
2.7
follows
immediately
—
i.e.,
by
“specializing
functions
to
points”
—
from
the
definitions;
Theorem
2.5,
(ii);
Proposition
2.1,
(iii);
Remark
2.1.1;
and
[Tama],
Lemma
4.7.
Here,
we
note
that,
in
the
present
situation,
the
isomorphism
∧
×
×
∧
∼
·
(k
X
)
→
K
Y
×
·
(k
Y
×
)
K
X
∼
×
of
Theorem
2.5,
(ii),
necessarily
induces
an
isomorphism
K
X
→
K
Y
×
[cf.
the
as-
sumption
that
Σ
†
=
Primes
†
].
Indeed,
this
is
immediate
in
the
finite
field
case.
In
the
nonarchimedean
local
field
case,
it
follows
via
the
arguments
applied
in
the
proof
of
[Mzk8],
Theorem
4.10:
That
is
to
say,
we
assume
for
simplicity
that
S
⊆
X(k
X
);
×
then
if
f
∈
K
X
,
and
x
∈
S
is
a
point
that
does
not
lie
in
the
divisor
of
zeroes
and
poles
of
f
,
then
let
us
observe
that
the
subset
×
×
⊆
f
·
(k
X
)
f
·
k
X
∧
may
be
characterized
as
the
subset
of
elements
whose
values
[cf.
Proposition
2.1,
×
×
∧
⊆
(k
X
)
.
Note
that
since,
for
a
given
x
1
∈
S,
there
clearly
exist
(iii)]
at
x
lie
in
k
X
×
f
∈
K
X
[at
least
after
possibly
passing
to
an
appropriate
connected
finite
étale
covering
of
X]
that
have
a
zero
or
pole
at
x
1
but
not
at
some
other
x
∈
S,
this
observation
allows
us
to
recover
the
canonical
discrete
structure
[cf.
[Mzk8],
Defi-
nition
4.1,
(iii);
the
proof
of
[Mzk8],
Theorem
4.10]
on
the
decomposition
groups
in
cl
Π
c-ab
U
S
[where
S
1
⊆
X
is
an
arbitrary
finite
subset
containing
S,
which
corresponds,
1
42
SHINICHI
MOCHIZUKI
say,
to
a
subset
T
1
⊆
Y
cl
that
contains
T
]
at
arbitrary
points
[i.e.,
arbitrary
“x
1
”]
of
S.
Thus,
by
applying
this
canonical
discrete
structure
[as
in
the
proof
of
[Mzk8],
Theorem
4.10],
we
may
recover
the
subset
×
×
f
·
k
X
⊆
f
·
(k
X
)
∧
×
for
arbitrary
f
∈
K
X
[i.e.,
even
f
that
have
a
zero
or
pole
at
every
point
of
S]
as
the
subset
of
elements
for
which
the
restriction
to
each
point
x
of
S
either
lies
in
×
×
∧
k
X
⊆
(k
X
)
or
[when
the
element
in
question
has
a
zero
or
pole
at
x]
is
compatible
with
the
canonical
discrete
structure
at
x.
Since
this
characterization
of
the
subset
×
×
∧
⊆
f
·
(k
X
)
is
manifestly
compatible
[in
light
of
the
Green-compatibility
f
·
k
X
∼
c-ab
assumption
on
α]
with
the
isomorphisms
Π
c-ab
U
S
1
→
Π
V
T
1
induced
by
α,
we
thus
conclude
that
the
isomorphism
∧
∼
∧
×
×
K
X
·
(k
X
)
→
K
Y
×
·
(k
Y
×
)
∧
×
×
×
of
Theorem
2.5,
(ii),
maps
the
subset
K
X
⊆
K
X
·
(k
X
)
onto
the
subset
K
Y
×
⊆
∧
K
Y
×
·
(k
Y
×
)
,
as
desired.
Remark
2.7.1.
Suppose,
in
the
situation
of
Corollary
2.7,
that
S
=
X
cl
,
T
=
Y
cl
.
Then
unlike
the
situation
discussed
in
[Tama],
one
has
the
freedom
to
evaluate
functions
at
arbitrary
points
of
the
entire
sets
X
cl
,
Y
cl
,
as
opposed
to
just
certain
restricted
subsets
S
⊆
X
cl
,
T
⊆
Y
cl
.
Thus,
instead
of
applying
[Tama],
Lemma
4.7,
one
may
instead
apply
the
somewhat
easier
argument
implicit
in
[Uchi],
§3,
Lemmas
8-11
[which
is
used
to
treat
the
function
field
case].
Thus,
in
light
of
Remark
2.6.1
[together
with
the
portion
of
Theorem
1.16,
(i),
concerning
the
preservation
of
decomposition
groups
of
cusps],
Corollary
2.7
implies
the
following
result,
in
the
affine
case:
Corollary
2.8.
(Point-theoretic
Isomorphisms
in
the
Affine
Case)
Let
U
,
V
be
affine
hyperbolic
curves
over
a
finite
or
nonarchimedean
local
field.
Suppose
that
Σ
=
Primes.
Write
Δ
U
(respectively,
Δ
V
)
for
the
maximal
cuspidally
pro-Σ
†
quotient
of
the
maximal
pro-Σ
quotient
of
the
tame
geo-
metric
fundamental
group
of
U
(respectively,
V
)
[where
“tame”
is
with
respect
to
the
complement
of
U
(respectively,
V
)
in
its
canonical
compactification],
and
Π
U
(respectively,
Π
V
)
for
the
corresponding
quotient
of
the
étale
fundamental
group
of
U
(respectively,
V
).
Then
any
point-theoretic
isomorphism
∼
β
:
Π
U
→
Π
V
arises
from
a
uniquely
determined
commutative
diagram
of
schemes
∼
U
⏐
⏐
→
U
→
∼
V
⏐
⏐
V
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
43
in
which
the
horizontal
arrows
are
isomorphisms;
the
vertical
arrows
are
the
pro-
finite
étale
coverings
determined
by
the
profinite
groups
Π
U
,
Π
V
.
Remark
2.8.1.
In
light
of
the
results
of
[Tama]
[cf.
Remarks
1.18.1,
1.18.2],
Corollary
2.8
is
only
truly
of
interest
in
the
case
of
nonarchimedean
local
fields.
Definition
2.9.
Suppose
that
k
is
a
nonarchimedean
local
field.
(i)
A
[necessarily
affine]
hyperbolic
curve
U
over
k
will
be
said
to
be
of
strictly
Belyi
type
if
it
is
defined
over
a
number
field
and
isogenous
[cf.
§0]
to
a
hyperbolic
curve
of
genus
zero.
(ii)
A
[necessarily
affine]
hyperbolic
curve
U
over
k
will
be
said
to
be
of
Belyi
type
if
it
is
defined
over
a
number
field,
and,
moreover,
for
some
positive
integer
m,
there
exists
a
finite
sequence
U
=
U
1
U
2
.
.
.
U
m−1
U
m
of
hyperbolic
orbicurves
[cf.
§0]
U
j
such
that
U
m
is
a
tripod
[cf.
§0],
and,
moreover,
for
each
j
=
1,
.
.
.
,
m
−
1,
U
j+1
is
related
to
U
j
in
one
of
the
following
ways:
(a)
there
exists
a
finite
étale
morphism
U
j+1
→
U
j
[i.e.,
“U
j+1
is
a
finite
étale
covering
of
U
j
”];
(b)
there
exists
a
finite
étale
morphism
U
j
→
U
j+1
[i.e.,
“U
j+1
is
a
finite
étale
quotient
of
U
j
”];
(c)
there
exists
an
open
immersion
U
j
→
U
j+1
[i.e.,
in
the
terminology
of
[Mzk8],
“U
j+1
is
a
[hyperbolic]
partial
compactification
of
U
j
”];
(d)
there
exists
a
partial
coarsification
morphism
[cf.
§0]
U
j
→
U
j+1
[i.e.,
“U
j+1
is
a
partial
coarsification
of
U
j
”].
(iii)
A
[necessarily
affine]
hyperbolic
curve
U
over
k
will
be
said
to
be
of
quasi-
Belyi
type
if
it
is
defined
over
a
number
field
and
admits
a
connected
finite
étale
covering
V
→
U
such
that
V
admits
a
[not
necessarily
finite
or
étale!]
dominant
morphism
V
→
W
to
a
tripod
W
.
Remark
2.9.1.
It
is
immediate
that
every
hyperbolic
curve
of
strictly
Belyi
type
is
also
of
Belyi
type
[as
the
terminology
suggests].
Moreover,
one
verifies
easily
by
“induction
on
m”
[where
“m”
is
as
in
Definition
2.9,
(ii)]
that
every
hyperbolic
curve
of
Belyi
type
is
also
of
quasi-Belyi
type
[as
the
terminology
suggests].
It
is
not
difficult
to
see
that
there
exist
[multiply]
punctured
elliptic
curves
that
are
of
Belyi
type,
but
not
of
strictly
Belyi
type
[cf.
Remark
2.13.2
below].
On
the
other
hand,
it
is
not
clear
to
the
author
at
the
time
of
writing
whether
or
not
there
exist
hyperbolic
curves
of
quasi-Belyi
type
that
are
not
of
Belyi
type.
44
SHINICHI
MOCHIZUKI
Remark
2.9.2.
Hyperbolic
curves
of
strictly
Belyi
type
are
precisely
the
sort
of
curves
considered
in
[Mzk8],
Corollaries
2.8,
3.2.
Remark
2.9.3.
The
author
would
like
to
thank
A.
Tamagawa
for
useful
discus-
sions
concerning
Definition
2.9,
(ii),
especially
Definition
2.9,
(ii),
(d).
Proposition
2.10.
(Decomposition
Groups
of
Curves
of
Quasi-Belyi
Type)
Let
U
(respectively,
V
)
be
a
hyperbolic
curve
over
a
nonarchimedean
local
field.
Denote
the
base
field
of
U
(respectively,
V
)
by
k
U
(respectively,
k
V
),
the
étale
fundamental
group
of
U
(respectively,
V
)
by
Π
U
(respectively,
Π
V
)
[i.e.,
“we
take
Σ
=
Primes”].
Let
∼
β
:
Π
U
→
Π
V
be
an
isomorphism
of
profinite
groups.
Then:
(i)
If
U
is
of
quasi-Belyi
type,
then
the
closed
points
of
“DLoc-type”
[in
the
sense
of
[Mzk8],
Definition
2.4]
are
p
U
-adically
dense
[where
p
U
is
the
residue
characteristic
of
k
U
]
in
U
(k
U
).
(ii)
If
U
is
of
quasi-Belyi
type,
then
β
maps
every
decomposition
group
of
a
closed
point
of
U
isomorphically
onto
a
decomposition
group
of
a
closed
point
of
V
.
(iii)
If
both
U
,
V
are
of
quasi-Belyi
type,
then
β
is
point-theoretic.
(iv)
If
U
is
of
Belyi
type,
then
so
is
V
.
Proof.
The
proof
of
assertion
(i)
is
similar
to
the
proof
of
[Mzk8],
Corollary
2.8:
That
is
to
say,
in
the
terminology
of
loc.
cit.,
it
follows
formally
from
the
fact
that
U
is
of
quasi-Belyi
type
that
the
“algebraic”
closed
points
[i.e.,
closed
points
defined
over
a
number
field,
which
are
manifestly
p
U
-adically
dense
in
U
(k
U
)]
of
U
are
of
“DLoc-type”
[cf.
the
proof
of
[Mzk8],
Corollary
2.8]:
Indeed,
it
suffices
to
consider
the
following
commutative
diagram
of
hyperbolic
curves,
whose
existence
follows
from
the
assumption
that
U
is
of
quasi-Belyi
type:
U
←−
V
⏐
⏐
−→
W
⏐
⏐
V
−→
→
U
−→
U
W
Here,
the
“hooked
arrow
→”
is
an
open
immersion;
all
of
the
“non-hooked
arrows”
except
for
V
→
W
,
V
→
W
are
finite
étale
morphisms;
V
→
W
,
V
→
W
are
dominant;
the
finite
étale
morphism
U
→
U
is
obtained
by
a
base-change
to
a
finite
extension
of
the
base
field
k
U
;
and
W
is
a
tripod
[so
W
→
W
is
a
“Belyi
map”].
Note
that
the
composite
arrow
V
→
W
→
U
→
U
may
be
thought
of
as
an
arrow
in
the
category
DLoc
k
U
(U
)
of
[Mzk8],
§2.
Observe,
moreover,
that
the
arrow
W
→
U
may
be
chosen
to
have
arbitrarily
designated
algebraic
closed
points
in
the
complement
of
its
image.
Thus,
we
conclude
that
this
diagram
exhibits
the
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
45
[arbitrarily
designated]
algebraic
closed
points
in
the
complement
of
the
image
of
W
→
U
→
U
as
points
of
DLoc-type,
as
desired.
This
completes
the
proof
of
assertion
(i).
In
light
of
assertion
(i)
[applied
to
the
various
connected
finite
étale
coverings
of
U
],
the
proof
of
assertion
(ii)
is
entirely
similar
to
the
proof
of
[Mzk8],
Corollary
3.2:
That
is
to
say,
by
[Mzk8],
Corollary
2.5,
it
follows
that
β
maps
decomposition
groups
of
DLoc-type
of
U
to
decomposition
groups
of
DLoc-type
of
V
.
Thus,
assertion
(ii)
follows
by
applying
[Mzk8],
Lemma
3.1
[where
the
density
statement
of
assertion
(i)
concerning
points
of
DLoc-type
allows
one
to
replace
the
“algebraicity”
condition
of
[Mzk8],
Lemma
3.1,
(iii),
by
the
condition
that
the
points
in
question
be
of
DLoc-
type].
Finally,
assertion
(iii)
follows
formally
from
assertion
(ii)
[and
Proposition
2.3,
(ii)].
Finally,
we
consider
assertion
(iv).
First,
I
claim
that
by
applying
the
iso-
morphism
β
[and
thinking
of
hyperbolic
orbicurves
as
being
represented
by
their
associated
étale
fundamental
groups],
one
may
transform
the
sequence
U
=
U
1
U
2
.
.
.
U
m−1
U
m
of
Definition
2.9,
(ii),
into
a
sequence
V
=
V
1
V
2
.
.
.
V
m−1
V
m
that
also
satisfies
the
conditions
of
Definition
2.9,
(ii),
in
such
a
way
that
we
also
∼
obtain
compatible
isomorphisms
β
j
:
Π
U
j
→
Π
V
j
[where
j
=
1,
.
.
.
,
m;
β
1
=
β].
Indeed,
we
reason
by
induction
on
m.
If
[for
j
=
1,
.
.
.
,
m
−
1]
U
j+1
is
related
to
U
j
as
in
(a)
[of
Definition
2.9,
(ii)],
then
it
is
immediate
[by
thinking
in
terms
of
open
subgroups
of
Π
U
j
,
Π
V
j
]
that
one
may
construct
[from
V
j
]
a
V
j+1
related
to
V
j
as
in
(a).
If
U
j+1
is
related
to
U
j
as
in
(b)
(respectively,
(c)),
then
it
follows
from
[Mzk6],
Theorem
2.4
(respectively,
[Mzk8],
Theorem
1.3,
(iii)
[cf.
also
[Mzk8],
Theorem
2.3]),
that
one
may
construct
[from
V
j
]
a
V
j+1
related
to
V
j
as
in
(b)
(respectively,
(c)).
If
U
j+1
is
related
to
U
j
as
in
(d),
then
Π
U
j+1
is
obtained
from
Π
U
j
by
forming
the
quotient
of
Π
U
j
by
the
closed
normal
subgroup
of
Π
U
j
generated
by
some
finite
collection
of
elements
of
Δ
U
j
that
belong
to
the
decomposition
groups
of
points
of
U
j
in
Δ
U
j
.
Thus,
by
Lemma
2.11,
(v),
below,
we
conclude
that
the
quotient
Π
U
j
Π
U
j+1
determines
a
quotient
Π
V
j
Π
V
j+1
that
corresponds
to
a
partial
coarsification
V
j
→
V
j+1
,
as
desired.
Finally,
if
U
m
is
a
tripod,
the
existence
∼
of
the
isomorphism
Π
U
m
→
Π
V
m
implies
that
V
m
is
also
a
tripod
[cf.
[Mzk5],
Lemma
1.3.9].
This
completes
the
proof
of
the
claim.
Thus,
to
complete
the
proof
of
assertion
(iv),
it
suffices
to
verify
that
V
is
defined
over
a
number
field.
But
observe
that
since
U
is
defined
over
a
number
field,
there
exists
a
diagram
of
hyperbolic
curves
[i.e.,
in
essence,
a
“Belyi
map”]
U
m
←−
U
m
→
U
−→
U
where
the
“hooked
arrow
→”
is
an
open
immersion;
the
“non-hooked
arrows”
are
finite
étale
morphisms;
and
the
finite
étale
morphism
U
→
U
is
obtained
by
46
SHINICHI
MOCHIZUKI
a
base-change
to
a
finite
extension
of
the
base
field
k
U
.
Now
the
isomorphisms
∼
∼
Π
U
m
→
Π
V
m
,
Π
U
→
Π
V
allow
us
to
transform
[cf.
[Mzk8],
Theorem
2.3
and
its
proof]
this
diagram
into
a
similar
diagram
V
m
←−
V
m
→
V
−→
V
whose
existence
[since
V
m
is
also
a
tripod!]
shows
that
V
is
also
defined
over
a
number
field,
as
desired.
This
completes
the
proof
of
assertion
(iv).
Remark
2.10.1.
Note
that
the
essential
reason
that
the
author
is
unable
to
prove
the
stronger
statement
of
Proposition
2.10,
(iv),
in
the
quasi-Belyi
case
is
that,
in
the
notation
of
the
proof
of
Proposition
2.10,
(i),
it
is
unclear
how
to
construct
[at
the
level
of
arithmetic
fundamental
groups]
the
dominant
morphism
V
→
W
from
V
.
That
is
to
say,
unlike
the
situation
involving
the
operations
of
Definition
2.9,
(ii),
(a),
(b),
(c),
(d),
it
is
by
no
means
clear
how
to
construct,
via
purely
group-
theoretic
operations,
the
quotient
of
an
arithmetic
fundamental
group
arising
from
an
arbitrary
dominant
morphism.
Lemma
2.11.
(Finite
Subgroups
of
Fundamental
Groups
of
Hyperbolic
Orbicurves)
Let
W
be
a
hyperbolic
orbicurve
over
an
algebraically
closed
field
of
characteristic
zero;
Σ
W
a
nonempty
set
of
prime
numbers.
Denote
the
maximal
pro-Σ
W
quotient
of
the
étale
fundamental
group
of
W
by
Δ
W
;
suppose
that
W
admits
a
finite
étale
covering
by
a
hyperbolic
curve
that
arises
from
an
open
subgroup
of
Δ
W
.
Let
A
⊆
Δ
W
(respectively,
B
⊆
Δ
W
)
be
the
decomposition
group
[well-defined
up
to
conjugation
in
Δ
W
]
of
a
closed
point
w
A
(respectively,
w
B
)
of
W
;
suppose
that
w
A
=
w
B
.
Then:
(i)
A,
B
are
cyclic.
(ii)
A
Δ
W
.
B
=
{1}.
In
particular,
if
A
=
{1},
then
A
is
normally
terminal
in
(iii)
The
order
of
every
finite
cyclic
closed
subgroup
C
⊆
Δ
W
divides
the
order
of
W
[cf.
§0].
(iv)
Every
finite
nontrivial
closed
subgroup
C
⊆
Δ
W
is
contained
in
a
decomposition
group
of
a
unique
closed
point
of
W
.
(v)
The
nontrivial
decomposition
groups
of
closed
points
of
W
may
be
charac-
terized
as
the
maximal
finite
nontrivial
closed
subgroups
of
Δ
W
.
Proof.
Assertion
(i)
follows
immediately
from
the
well-known
[and
easily
verified]
fact
that
the
absolute
Galois
group
of
a
complete
discrete
valuation
field
with
algebraically
closed
residue
field
of
characteristic
zero
is
cyclic.
Next,
we
consider
assertion
(ii).
Let
C
⊆
A
B
be
a
subgroup
of
prime
order
l
∈
Σ
W
.
Now
consider
a
normal
open
subgroup
H
⊆
Δ
W
such
that
the
covering
W
H
→
W
determined
by
H
is
a
hyperbolic
curve.
Note
that
this
implies
that
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
47
A
H
=
B
H
=
C
H
=
{1}
[cf.,
e.g.,
assertion
(iii),
which
will
be
proven
below
without
applying
the
present
assertion
(ii)].
Write
W
H
→
W
C
→
W
for
the
covering
determined
by
the
open
subgroup
C
·
H
⊆
Δ
W
.
Observe
that
there
exist
,
w
B
of
W
C
that
lift
w
A
,
w
B
,
respectively,
and
whose
decomposition
closed
points
w
A
groups
[well-defined
up
to
conjugation
in
C
·
H]
are
equal
to
C.
Note
that
since
W
H
is
a
hyperbolic
curve,
and
C
is
of
prime
order
l,
it
follows
that
the
order
of
every
closed
point
of
W
C
is
equal
to
either
1
or
l.
Now
if
W
C
is
affine,
then
let
v
be
a
cusp
of
W
C
.
If
W
C
is
proper
and
admits
≥
3
points
of
order
l,
then
let
v
be
a
point
of
W
C
of
order
l
such
that
v
=
w
A
,
w
B
.
Note
that
if
W
C
is
proper
and
admits
≤
2
points
of
order
l,
then
it
follows
from
the
hyperbolicity
assumption
that
the
coarsification
of
W
C
is
a
proper
smooth
curve
of
genus
≥
1;
thus,
by
replacing
H
by
an
appropriate
open
subgroup
of
H,
one
verifies
immediately
that
one
may
assume
without
loss
of
generality
that
either
W
C
is
affine
or
W
C
admits
≥
3
points
of
order
l.
Now
observe
that
W
C
admits
a
finite
étale
cyclic
covering
W
C
→
W
C
of
degree
l
which
is
étale
over
the
compactification
of
the
coarsification
of
W
C
,
except
over
the
,
points
in
the
compactification
of
the
coarsification
of
W
C
corresponding
to
v,
w
B
over
which
W
C
is
totally
ramified.
In
particular,
it
follows
that
any
point
of
W
C
lying
over
w
A
(respectively,
w
B
)
is
of
order
l
(respectively,
1),
thus
contradicting
the
observation
that
the
decomposition
groups
[well-defined
up
to
conjugation
in
C
·
H]
of
w
A
,
w
B
are
equal
to
C.
This
completes
the
proof
that
A
B
=
{1}.
By
applying
this
fact
to
arbitrary
finite
étale
coverings
of
W
,
it
follows
formally
[cf.
Proposition
2.6,
(ii)]
that
A
is
normally
terminal
in
Δ
W
,
whenever
A
=
{1}.
Next,
we
consider
assertion
(iii).
Denote
the
order
of
W
by
n.
Now
if
C
⊆
Δ
W
is
a
nontrivial
finite
cyclic
closed
subgroup,
then
there
exists
a
normal
open
subgroup
def
N
⊆
Δ
W
such
that
C
N
=
{1}.
In
particular,
it
follows
that
if
we
take
H
=
C
·N
[so
H
⊆
Δ
W
is
an
open
subgroup],
then
the
natural
map
C
→
H
ab
is
injective.
On
the
other
hand,
if
we
denote
by
W
H
→
W
the
covering
determined
by
H,
then
it
is
clear
that
the
order
of
W
H
divides
n,
hence
that
H
ab
is
an
extension
of
a
torsion-
free
profinite
abelian
group
by
a
finite
abelian
group
annihilated
by
n.
Thus,
we
conclude
from
the
injection
C
→
H
ab
that
the
order
of
C
divides
n,
as
desired.
This
completes
the
proof
of
assertion
(iii).
Next,
we
consider
assertion
(iv).
First,
let
us
observe
that
uniqueness
follows
formally
from
assertion
(ii).
Next,
let
us
verify
assertion
(iv)
under
the
further
assumption
that
C
is
solvable.
By
induction
on
the
order
of
C,
we
may
assume
that
[at
least]
one
of
the
following
conditions
is
satisfied:
(a)
C
is
an
extension
of
a
group
of
prime
order
by
a
nontrivial
subgroup
C
1
⊆
C
which
is
contained
in
the
decomposition
group
A;
(b)
C
is
of
prime
order
l
∈
Σ
W
.
If
(a)
is
satisfied,
then
by
replacing
W
by
a
finite
étale
covering
of
W
determined
by
a
suitable
open
subgroup
containing
C,
we
may
assume
that
(C
1
⊆)
A
⊆
C.
Thus,
if
A
=
C,
then
A
=
C
1
is
normal
in
C.
But
this
implies,
by
the
normal
terminality
portion
of
assertion
(ii),
that
A
=
C,
a
contradiction.
Thus,
(a)
implies
that
C
⊆
A.
If
(b)
is
satisfied,
then
we
argue
as
follows:
Observe
that
by
assertion
(iii),
every
open
subgroup
H
⊆
Δ
W
that
contains
C
determines
a
finite
étale
covering
W
H
→
W
such
that
the
order
of
W
H
is
divisible
by
l.
Write
Stack
l
(W
H
)
for
the
set
of
closed
points
of
W
H
whose
order
is
divisible
by
l.
Now
observe
that
48
SHINICHI
MOCHIZUKI
since
the
order
of
W
H
is
divisible
by
the
prime
number
l,
it
follows
that
Stack
l
(W
H
)
is
nonempty.
Since
the
set
Stack
l
(W
H
)
is
finite
and
nonempty,
we
thus
conclude
that,
if
we
allow
H
to
vary
[among
open
subgroups
H
⊆
Δ
W
that
contain
C],
then
the
inverse
limit
lim
←−
Stack
l
(W
H
)
H
is
nonempty.
But,
unraveling
the
definitions,
this
means
precisely
that
C
contains
the
decomposition
group
D
associated
to
some
compatible
system
of
points
of
the
sets
Stack
l
(W
H
).
Since
D
is
of
order
divisible
by
l,
we
thus
conclude
that
D
=
C,
as
desired.
This
completes
the
proof
of
assertion
(iv)
for
C
solvable.
On
the
other
hand,
a
well-known
theorem
from
the
theory
of
finite
groups
asserts
that
a
finite
group
in
which
every
Sylow
subgroup
is
cyclic
is
solvable
[cf.
[Scott],
p.
356].
Thus,
in
light
of
assertion
(i),
we
conclude
that
assertion
(iv)
for
C
solvable
implies
assertion
(iv)
for
C
arbitrary.
Finally,
we
observe
that
assertion
(v)
follows
formally
from
assertions
(ii),
(iv).
Remark
2.11.1.
The
author
would
like
to
thank
A.
Tamagawa
for
informing
him
of
Lemma
2.11
and,
in
particular,
of
the
theorem
on
finite
groups
that
was
applied
in
the
proof
of
Lemma
2.11,
(iv).
We
are
now
ready
to
state
the
following
“absolute
p-adic
version
of
the
Grothen-
dieck
Conjecture”
for
hyperbolic
curves
of
Belyi
or
quasi-Belyi
type:
Corollary
2.12.
(Curves
of
Belyi
or
Quasi-Belyi
Type)
Let
U
(respectively,
V
)
be
a
hyperbolic
curve
over
a
nonarchimedean
local
field.
Denote
the
base
field
of
U
(respectively,
V
)
by
k
U
(respectively,
k
V
),
the
étale
fundamental
group
of
U
(respectively,
V
)
by
Π
U
(respectively,
Π
V
)
[i.e.,
“we
take
Σ
=
Primes”].
Suppose
further
that
at
least
one
of
the
following
conditions
holds:
(a)
both
U
and
V
are
of
quasi-Belyi
type;
(b)
either
U
or
V
[but
not
necessarily
both!]
is
of
Belyi
type.
Then
any
isomorphism
of
profinite
groups
∼
β
:
Π
U
→
Π
V
arises
from
a
uniquely
determined
commutative
diagram
of
schemes
∼
U
⏐
⏐
→
U
→
∼
V
⏐
⏐
V
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
49
in
which
the
horizontal
arrows
are
isomorphisms;
the
vertical
arrows
are
the
pro-
finite
étale
coverings
determined
by
the
profinite
groups
Π
U
,
Π
V
.
Proof.
In
light
of
Proposition
2.10,
(iii),
(iv)
[cf.
also
Remark
2.9.1],
Corollary
2.12
follows
formally
from
Corollary
2.8.
Remark
2.12.1.
Note
that
in
the
proof
of
Proposition
2.10,
Corollary
2.12,
it
is
necessary,
in
the
quasi-Belyi
case,
to
apply
the
full
“Hom
version”
of
[Mzk4],
Theorem
A.
This
differs
from
the
situation
of
[Mzk8],
Corollaries
2.8,
3.2
—
i.e.,
where
one
only
treats
hyperbolic
curves
of
strictly
Belyi
type
—
or,
indeed,
of
the
portion
of
Proposition
2.10,
Corollary
2.12,
that
concerns
curves
of
Belyi
type,
in
which
the
“isomorphism
version”
of
[Mzk4],
Theorem
A,
suffices
[cf.
[Mzk8],
Remark
2.8.1].
Thus,
in
the
terminology
of
[Mzk6],
Definition
3.7,
the
portion
of
Corollary
2.12
concerning
hyperbolic
curves
of
Belyi
type
admits
the
following
formal
consequence:
Corollary
2.13.
(Absoluteness
of
Curves
of
Belyi
Type)
Every
hyperbolic
curve
of
Belyi
type
over
a
nonarchimedean
local
field
is
absolute.
Remark
2.13.1.
It
is
interesting
to
note
that
the
essential
property
that
underlies
the
absoluteness
of
Corollary
2.13
is
the
existence
of
a
Belyi
map
[since
the
curve
is
defined
over
a
number
field],
which,
in
the
context
of
the
theory
of
[Mzk8],
§2,
may
be
regarded
as
a
sort
of
endomorphism
of
the
curve.
From
this
point
of
view,
Corollary
2.13
is
reminiscent
of
[Mzk6],
Corollary
3.8,
which
states
that
the
“canonical
curves”
of
p-adic
Teichmüller
theory
are
absolute.
Indeed,
from
the
point
of
view
of
the
theory
of
[Mzk2],
this
canonicality
may
be
regarded
as
the
existence
of
a
sort
of
“Frobenius
endomorphism”
of
the
curve.
It
is
also
interesting
to
note
that
both
of
these
results
assert
that
every
member
of
some
countable
collection
of
nonarchimedean
hyperbolic
curves
is
absolute.
Remark
2.13.2.
In
the
context
of
Remark
2.13.1,
it
is
interesting
to
note
that,
unlike
the
canonical
curves
discussed
in
[Mzk6],
§3,
the
set
of
points
determined
by
the
hyperbolic
curves
of
strictly
Belyi
type
fails,
for
all
pairs
(g,
r)
such
that
2g
−
2
+
r
≥
3,
g
≥
1,
to
be
Zariski
dense
in
the
moduli
stack
of
hyperbolic
curves
of
type
(g,
r).
Indeed,
this
follows
immediately
from
[Mzk1],
Theorem
B.
On
the
other
hand,
it
is
not
clear
to
the
author
at
the
time
of
writing
whether
or
not
the
set
of
points
determined
by
the
hyperbolic
curves
of
Belyi
(respectively,
quasi-Belyi)
type
is
Zariski
dense
in
the
moduli
stack
of
hyperbolic
curves
of
type
(g,
r)
[when,
say,
2g
−
2
+
r
≥
3,
g
≥
2].
Note,
however,
that
when
g
=
0,
1,
[one
verifies
easily
that]
every
hyperbolic
curve
of
type
(g,
r)
that
is
defined
over
a
number
field
is
automatically
of
Belyi
type.
50
SHINICHI
MOCHIZUKI
Section
3:
Maximal
Pro-l
Cuspidalizations
In
this
§,
we
apply
the
theory
of
the
weight
filtration
[cf.
[Kane],
[Mtm]],
together
with
various
generalities
concerning
free
Lie
algebras
[cf.
the
Appendix],
to
construct,
in
the
finite
field
case,
“maximal
cuspidally
pro-l
cuspidalizations”
[cf.
Theorem
3.10],
whose
existence
implies,
under
quite
general
conditions
[cf.
Corollary
3.11
below],
that
an
isomorphism
“α”
as
in
Theorem
1.16,
(iii),
is
always
totally
globally
Green-compatible.
In
the
following
discussion,
we
maintain
the
notation
of
§2,
and
assume
further
throughout
the
present
§3
that
we
are
in
the
finite
field
case.
Definition
3.1.
Let
l
be
a
prime
number;
G,
H,
A
topologically
finitely
generated
pro-l
groups;
φ
:
H
→
A
a
[continuous]
homomorphism.
Suppose
further
that
A
is
abelian,
and
that
G
is
an
l-adic
Lie
group
[cf.,
e.g.,
[Serre],
Chapter
V,
§7,
§9,
for
basic
facts
concerning
l-adic
Lie
groups].
(i)
We
shall
refer
to
as
the
φ-central
filtration
on
H
the
filtration
defined
as
follows:
def
H(1)
=
H
def
H(2)
=
Ker(φ)
def
H(m)
=
the
subgroup
topologically
generated
by
the
commutators
[H(a),
H(b)],
where
a
+
b
=
m,
∀
m
≥
3
Thus,
in
words,
this
filtration
on
H
is
the
“fastest
decreasing
central
filtration
among
those
central
filtrations
whose
top
quotient
factors
through
φ”.
We
shall
say
that
H
is
φ-nilpotent
if
H(m)
=
{1}
for
sufficiently
large
m.
If
H
is
φ-nilpotent
when
φ
is
taken
to
be
the
natural
surjection
H
H
ab
to
its
abelianization
H
ab
,
then
we
shall
say
that
H
is
nilpotent.
In
the
following,
for
a,
b,
n
∈
Z
such
that
1
≤
a
≤
b,
n
≥
1,
we
shall
write
def
H(a/b)
=
H(a)/H(b)
and
def
Gr(H)(n)
=
def
H(m/m
+
1)
⊆
Gr(H)
=
Gr(H)(1)
m≥n
def
Gr(H)(a/b)
=
Gr(H)(a)/Gr(H)(b)
and
append
a
subscript
Q
l
(respectively,
F
l
)
to
these
objects
to
denote
the
result
of
tensoring
over
Z
l
with
Q
l
(respectively,
F
l
).
Thus,
Gr(H),
Gr
Q
l
(H),
Gr
F
l
(H)
are
graded
Lie
algebras
over
Z
l
,
Q
l
,
F
l
,
respectively;
Gr(H)(n)
⊆
Gr(H)
is
a
[Lie
algebra-theoretic]
ideal.
Also,
if
Z
a
≥
1,
then
we
shall
write:
def
H(a/∞)
=
←
lim
−
H(a/b)
b
[where
b
ranges
over
the
integers
≥
a
+
1].
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
51
(ii)
We
shall
denote
by
Lie(G)
the
Lie
algebra
over
Q
l
determined
by
G.
If
G
is
nilpotent,
then
Lie(G)
is
a
nilpotent
Lie
algebra
over
Q
l
,
hence
determines
a
connected,
unipotent
linear
algebraic
group
Lin(G),
which
we
shall
refer
to
as
the
linear
algebraic
group
associated
to
G.
In
this
situation,
there
exists
[cf.,
e.g.,
Remark
3.3.2
below]
a
natural
[continuous]
homomorphism
[with
open
image]
G
→
Lin(G)(Q
l
)
[from
G
to
the
l-adic
Lie
group
determined
by
the
Q
l
-valued
points
of
Lin(G)]
which
is
uniquely
determined
[since
Lin(G)
is
connected
and
unipotent!]
by
the
condition
that
it
induce
the
identity
morphism
on
the
associated
Lie
algebras.
In
the
situation
of
(i),
if
Z
a
≥
1,
then
we
shall
write:
def
lim
Lie(H(a/∞))
=
←
−
Lie(H(a/b));
b
def
Lin(H(a/∞))
=
←
lim
−
Lin(H(a/b))
b
[where
b
ranges
over
the
integers
≥
a
+
1;
we
recall
that
it
is
well-known
[or
easily
verified]
that
each
H(a/b)
is
an
l-adic
Lie
group].
Now
let
us
fix
a
prime
number
l
∈
Σ
†
.
For
S
⊆
X(k)
a
finite
subset,
let
us
denote
by
(l)
(l)
Δ
U
S
Δ
U
S
;
Δ
X
Δ
X
the
maximal
pro-l
quotients
and
by
(l)
Π
U
S
Π
U
S
;
(l)
Π
X
Π
X
(l)
(l)
the
quotients
of
Π
U
S
,
Π
X
by
the
kernels
of
Δ
U
S
Δ
U
S
,
Δ
X
Δ
X
.
[Here,
we
recall
that
Δ
U
S
,
Π
U
S
are
as
defined
in
Proposition
1.8,
(ii),
(iii).]
Also,
for
x
∈
X
cl
,
let
us
write
(l)
(l)
D
x
(l)
[U
S
]
⊆
Π
U
S
;
I
x
(l)
[U
S
]
⊆
Δ
U
S
for
the
images
of
D
x
[U
S
],
I
x
[U
S
]
[notation
as
in
Proposition
2.1],
respectively,
in
(l)
Π
U
S
.
Note
that
we
have
a
natural
surjection:
(l)
(l)
(l)
Δ
U
S
Δ
X
(Δ
X
)
ab
(l)
The
cup
product
on
the
group
cohomology
of
Δ
X
determines
an
isomorphism
[cf.
Proposition
1.3,
(ii)]
(l)
(l)
∼
(l)
Hom((Δ
X
)
ab
,
M
X
)
→
(Δ
X
)
ab
(l)
def
[where
we
write
M
X
=
M
X
⊗
Z
l
],
hence
a
natural
G
k
-equivariant
injection
(l)
(l)
M
X
→
∧
2
(Δ
X
)
ab
(l)
whose
image
we
denote
by
I
cup
.
52
SHINICHI
MOCHIZUKI
Definition
3.2.
We
shall
refer
to
the
central
filtration
(l)
{Δ
U
S
(m)}
(l)
(l)
(l)
on
Δ
U
S
with
respect
to
the
natural
surjection
Δ
U
S
(Δ
X
)
ab
as
the
weight
filtra-
(l)
tion
on
Δ
U
S
[cf.,
e.g.,
[Mtm],
§3,
p.
200].
def
Proposition
3.3.
(Freeness
and
Centralizers)
Let
x
∈
S.
Write
S
x
=
S\{x};
r
for
the
cardinality
of
S,
g
for
the
genus
of
X.
For
x
∈
S,
let
ζ
x
be
a
(l)
generator
of
I
x
[U
S
].
By
abuse
of
notation,
we
shall
also
denote
by
ζ
x
the
image
(l)
of
ζ
x
in
Δ
U
S
(2/3).
Then:
(l)
(i)
Gr(Δ
U
S
)
is
a
free
Lie
algebra
over
Z
l
[hence,
in
particular,
is
torsion-free
as
a
Z
l
-module]
which
is
freely
generated
by
2g
elements
(l)
α
1
,
.
.
.
,
α
g
,
β
1
,
.
.
.
,
β
g
∈
Δ
U
S
(1/2)
together
with
the
ζ
x
∈
Δ
U
S
(2/3),
for
x
∈
S
x
.
Alternatively,
for
an
appropriate
(l)
(l)
choice
of
the
elements
ζ
x
,
Gr(Δ
U
S
)
is
the
quotient
of
the
free
Lie
algebra
generated
(l)
by
α
1
,
.
.
.
,
α
g
,
β
1
,
.
.
.
,
β
g
,
together
with
the
ζ
x
∈
Δ
U
S
(2/3),
for
x
∈
S,
by
the
single
relation:
g
ζ
x
+
[α
n
,
β
n
]
=
0
x
∈S
n=1
At
a
more
intrinsic
level,
this
relation
is
a
generator
of
the
image
of
the
natural
G
k
-equivariant
morphism
(l)
M
X
→
(l)
I
x
[U
S
]
(l)
⊕
I
cup
x
∈S
(l)
∼
(l)
(l)
∼
(l)
[determined
by
the
various
natural
isomorphisms
M
X
→
I
x
[U
S
],
M
X
→
I
cup
]],
(l)
whose
codomain
maps
to
Gr(Δ
U
S
)
via
the
natural
G
k
-equivariant
morphism
(l)
I
x
[U
S
]
x
∈S
(l)
(l)
⊕
I
cup
→
Δ
U
S
(2/3)
(l)
(l)
[determined
by
the
natural
inclusions
I
x
[U
S
]
→
Δ
U
S
(2/3)
and
the
bracket
opera-
(l)
(l)
tion
∧
2
(Δ
X
)
ab
→
Δ
U
S
(2/3)].
(ii)
Let
ξ
be
any
of
the
elements
α
1
,
.
.
.
,
α
g
,
β
1
,
.
.
.
,
β
g
;
ζ
x
,
where
x
∈
S
x
,
(l)
(l)
of
(i).
Then
the
centralizer
in
Gr
Q
l
(Δ
U
S
)
of
[the
image
of
]
ξ
[in
Gr
Q
l
(Δ
U
S
)]
is
(l)
equal
to
Q
l
·
ξ.
In
particular,
the
Lie
algebra
Gr
Q
l
(Δ
U
S
)
is
center-free.
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
53
(l)
(iii)
Let
ξ
be
as
in
(ii).
Then
for
m
≥
1,
the
centralizer
in
Δ
U
S
(1/m
+
2)
of
(l)
(l)
[the
image
of
]
ξ
[in
Δ
U
S
(1/m
+
2)]
is
contained
in
the
subgroup
of
Δ
U
S
(1/m
+
2)
(l)
generated
by
[the
image
of
]
ξ
and
Δ
U
S
(m/m
+
2).
(iv)
Let
S
∗
⊆
S
be
a
subset
of
S.
Write
(l)
(l)
New
S
∗
⊆
Gr(Δ
U
S
)
for
the
sub-Lie
algebra
over
Z
l
generated
by
the
image
of
the
restriction
(l)
(l)
(l)
I
x
[U
S
]
⊆
I
x
[U
S
]
→
Δ
U
S
(2/3)
x
∈S
∗
x
∈S
to
the
direct
summands
indexed
by
elements
of
S
∗
of
the
morphism
of
(i),
and
(l)
def
(l)
New
S
∗
(a)
=
Gr(Δ
U
S
)(a)
(l)
(l)
def
(l)
(l)
New
S
∗
;
New
S
∗
(a/b)
=
New
S
∗
(a)/New
S
∗
(b)
for
a,
b
∈
(l)
Z
such
that
1
≤
a
≤
b.
Then,
in
the
notation
of
(i),
New
S
∗
is
a
free
Lie
algebra
over
Z
l
generated
by
the
elements
ζ
x
,
for
x
∈
S
∗
.
Moreover,
the
[“new”
and
“co-new”]
Z
l
-modules
(l)
New
S
∗
(a/b);
def
(l)
(l)
(l)
Cnw
S
∗
(a/b)
=
Gr(Δ
U
S
)(a/b)/New
S
∗
(a/b)
tor,(l)
are
free.
In
the
following
discussion,
we
shall
write
New
S
∗
Q/Z.
def
(l)
(a/b)
=
New
S
∗
(a/b)⊗
Proof.
Assertion
(i)
(respectively,
(ii))
is,
in
essence,
the
content
of
[Kane],
Propo-
sition
1
(respectively,
Proposition
A.1,
(ii),
(iii)).
Assertion
(iii)
follows
formally
from
assertion
(ii).
Finally,
we
consider
assertion
(iv).
By
Proposition
A.1,
(iii),
it
follows
that
any
free
Lie
algebra
over
F
l
with
≥
2
generators
is
center-free.
Thus,
let
M
be
the
module
determined
by
any
faithful
representation
[e.g.,
when
the
car-
dinality
of
S
∗
is
≥
2,
the
adjoint
representation]
of
the
free
Lie
algebra
F
over
F
l
in
the
formal
generators
ζ
x
,
where
x
∈
S
∗
.
Now
observe
that
we
obtain
an
def
(l)
action
of
Gr
F
l
(Δ
U
S
)
on
M
=
M
⊕
M
as
follows:
We
let
α
2
,
.
.
.
α
g
;
β
2
,
.
.
.
β
g
;
ζ
x
,
def
where
x
∈
S
0
=
S\S
∗
,
act
by
multiplication
by
0
on
M
.
We
let
α
1
,
β
1
act
on
M
=
M
⊕
M
via
the
matrices
ζ
x
0
0
0
x
∈S
∗
;
−1
0
0
0
respectively.
Finally,
we
let
ζ
x
,
where
x
∈
S
∗
,
act
on
M
via
the
following
matrix:
0
ζ
x
0
−ζ
x
Thus,
[by
assertion
(i)]
M
determines
a
representation
of
Gr
F
l
(Δ
U
S
)
whose
re-
(l)
(l)
(l)
striction
to
the
image
of
New
S
∗
⊗
Z
l
F
l
in
Gr
F
l
(Δ
U
S
)
determines
[via
the
natural
54
SHINICHI
MOCHIZUKI
(l)
surjection
F
New
S
∗
⊗
Z
l
F
l
]
a
faithful
representation
of
F
.
Thus,
we
conclude
that
(l)
(l)
the
natural
surjection
F
New
S
∗
⊗
Z
l
F
l
is
an
isomorphism,
and
that
New
S
∗
⊗
Z
l
F
l
(l)
injects
into
Gr
F
l
(Δ
U
S
).
Assertion
(iv)
now
follows
formally.
Remark
3.3.1.
The
author
wishes
to
thank
A.
Tamagawa
for
pointing
out
to
him
the
content
of
Proposition
3.3,
(i).
Remark
3.3.2.
One
way
to
verify
the
existence
of
the
homomorphism
“G
→
Lin(G)(Q
l
)”
of
Definition
3.1,
(ii),
is
to
think
of
G
as
a
quotient
of
a
free
pro-l
group
of
finite
even
rank
F
,
whose
associated
“Gr
Q
l
(−)”
is
a
center-free
free
Lie
algebra
[cf.
Proposition
3.3,
(i),
(ii),
in
the
case
of
r
=
1],
hence
determines
an
[infinite-dimensional,
over
Q
l
]
faithful
[cf.
Proposition
3.3,
(iii)]
unipotent
represen-
tation
[i.e.,
the
adjoint
representation
—
cf.
the
proof
of
Proposition
3.3,
(iv)]
of
F
.
More
precisely,
by
Proposition
3.3,
(iii),
it
follows
that
there
exists
a
unipotent
linear
representation
ρ
F
:
F
→
GL(V
)
on
a
finite-dimensional
Q
l
-vector
space
V
such
that
Ker(ρ
F
)
⊆
Ker(F
G).
But
this
implies
that
F
G
factors
through
a
quotient
F
Q
G
such
that
Q
is
nilpotent
and
admits
an
injective
homo-
morphism
of
topological
groups
ρ
Q
:
Q
→
Q
alg
(Q
l
)
[induced
by
ρ
F
],
where
Q
alg
is
a
connected,
unipotent
algebraic
group
over
Q
l
,
such
that
ρ
Q
is
a
local
isomor-
phism,
and
Ker(ρ
Q
)
⊆
Ker(Q
G).
Thus,
ρ
Q
determines
a
structure
of
l-adic
Lie
group
on
Q
such
that
the
morphism
Lie(ρ
Q
)
induced
by
ρ
Q
on
Lie
algebras
is
an
isomorphism.
Moreover,
the
morphism
induced
by
Q
G
on
Lie
algebras
factors
through
Lie(ρ
Q
),
thus
determining
a
homomorphism
of
[connected,
unipotent]
al-
gebraic
groups
Q
alg
→
Lin(G)
such
that
the
resulting
composite
homomorphism
Q
→
Q
alg
(Q
l
)
→
Lin(G)(Q
l
)
factors
[cf.
the
induced
morphisms
on
Lie
algebras,
together
with
the
fact
that
Lin(G)(Q
l
)
has
no
torsion!]
though
G,
thus
yielding
a
homomorphism
G
→
Lin(G)(Q
l
),
as
desired.
Next,
let
us
fix
an
x
∗
∈
S,
as
well
as
a
choice
of
decomposition
group
D
x
∗
[U
S
]
⊆
Π
U
S
[i.e.,
among
the
various
Π
U
S
-conjugates
of
this
subgroup]
associated
to
x
∗
.
[Thus,
D
x
∗
[U
S
]
determines
a
specific
subgroup
[i.e.,
not
just
a
conjugacy
class
of
subgroups]
(l)
(l)
D
x
∗
[U
S
]
⊆
Π
U
S
.]
Recall
that
the
natural
exact
sequences
1
→
I
x
∗
[U
S
]
→
D
x
∗
[U
S
]
→
G
k
→
1;
1
→
I
x
(l)
[U
S
]
→
D
x
(l)
∗
[U
S
]
→
G
k
→
1
∗
split.
[Indeed,
extracting
roots
of
any
local
uniformizer
of
X
at
x
∗
determines
such
a
splitting
—
cf.,
e.g.,
the
discussion
at
the
beginning
of
[Mzk8],
§4.]
In
the
following
discussion,
we
shall
fix
a
splitting
G
k
→
D
x
∗
[U
S
]
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
55
of
this
exact
sequence.
Thus,
this
splitting
determines
a
natural
action
of
G
k
[by
(l)
conjugation]
on
Δ
U
S
,
hence
also
on
def
(l)
(l)
def
(l)
Lin
U
S
(a/b)
=
Lin(Δ
U
S
(a/b))(Q
l
);
(l)
Lie
U
S
(a/b)
=
Lie(Δ
U
S
(a/b))
(l)
Gr
Q
l
(Δ
U
S
)(a/b)
[where
a,
b
∈
Z;
1
≤
a
≤
b].
Write
F
k
∈
G
k
for
the
Frobenius
element
of
G
k
.
In
the
following,
we
shall
denote
the
cardinality
of
k
by
q
k
.
(Galois
Invariant
Splitting)
Let
a,
b
∈
Z,
1
≤
a
≤
b.
Proposition
3.4.
(l)
(i)
The
eigenvalues
of
the
action
of
F
k
on
Lie
U
S
(a/a
+
1)
are
algebraic
num-
a/2
bers
all
of
whose
complex
absolute
values
are
equal
to
q
k
[i.e.,
“of
weight
a”].
(ii)
There
is
a
unique
G
k
-equivariant
isomorphism
of
Lie
algebras
∼
(l)
(l)
Lie
U
S
(a/b)
→
Gr
Q
l
(Δ
U
S
)(a/b)
∼
(l)
(l)
which
induces
the
identity
isomorphism
Lie
U
S
(c/c
+
1)
→
Gr
Q
l
(Δ
U
S
)(c/c
+
1),
for
all
c
∈
Z
such
that
a
≤
c
≤
b
−
1.
(l)
(iii)
The
isomorphism
of
(ii)
together
with
the
natural
inclusions
I
x
[U
S
]
→
(l)
(l)
Δ
U
S
for
x
∈
S
[which
are
well-defined
up
to
Δ
U
S
-conjugation]
determine
a
G
k
-
equivariant
morphism
I
x
(l)
[U
S
]
⊗
Q
l
(l)
(l)
⊕
Lie
U
S
(1/2)
→
Lie
U
S
(1/∞)
x∈S
(l)
which
exhibits,
in
a
G
k
-equivariant
fashion,
Lie
U
S
(1/∞)
as
the
quotient
of
the
completion
[with
respect
to
the
filtration
topology]
of
the
free
Lie
algebra
generated
by
the
finite
dimensional
Q
l
-vector
space
I
x
(l)
[U
S
]
⊗
Q
l
(l)
⊕
Lie
U
S
(1/2)
x∈S
(l)
[equipped
with
a
natural
grading,
hence
also
a
filtration,
by
taking
the
I
x
[U
S
]
⊗
(l)
Q
l
to
be
of
weight
2,
Lie
U
S
(1/2)
to
be
of
weight
1],
by
the
single
relation
deter-
mined
by
the
image
of
the
morphism
(l)
M
X
⊗
Q
l
→
x∈S
(l)
I
x
(l)
[U
S
]
⊗
Q
l
⊕
(I
cup
⊗
Q
l
)
56
SHINICHI
MOCHIZUKI
of
Proposition
3.3,
(i),
tensored
with
Q
l
.
(l)
that
(l)
(iv)
For
each
g
∈
Lin
U
S
(1/∞),
there
exists
a
unique
h
∈
Lin
U
S
(1/∞)
such
F
k
◦
Inn
g
=
Inn
h
◦
F
k
◦
Inn
h
−1
(l)
[where
“Inn”
denotes
the
inner
automorphism
of
Lin
U
S
(1/∞)
defined
by
conjuga-
(l)
tion
by
the
subscripted
element].
Moreover,
when
g
lies
in
the
image
of
I
x
∗
⊗
Q
l
(l)
[which
is
stabilized
by
the
action
of
F
k
],
h
also
lies
in
the
image
of
I
x
∗
⊗
Q
l
.
Proof.
Assertion
(i)
follows
immediately
from
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
—
cf.,
e.g.,
[Mumf],
p.
206.
Assertion
(ii)
(respec-
tively,
(iii);
(iv))
follows
formally
from
assertion
(i)
(respectively,
and
Proposition
3.3,
(i);
and
successive
approximation
of
h
with
respect
to
the
natural
filtration
(l)
(l)
Lin
U
S
(a/∞)
⊆
Lin
U
S
(1/∞)).
Next,
let
S
∗
⊆
S
def
be
a
subset
such
that
x
∗
∈
S
∗
;
S
0
=
S\S
∗
.
In
the
following,
we
shall
regard
(l)
Lin
U
S
(a/b)
as
being
equipped
with
its
natural
l-adic
topology.
Thus,
G
k
acts
con-
(l)
(l)
tinuously
on
Lin
U
S
(a/b),
Lie
U
S
(a/b),
and
we
have
natural
G
k
-equivariant
surjec-
tions:
(l)
(l)
(l)
(l)
Lin
U
S
(a/b)
Lin
U
S
(a/b);
Lie
U
S
(a/b)
Lie
U
S
(a/b)
0
Let
us
write
0
(l)
Lin
U
S
/U
S
(a/b);
0
(l)
Lie
U
S
/U
S
(a/b)
0
for
the
kernels
of
these
surjections.
In
the
following,
to
simplify
the
notation,
we
shall
often
omit
the
superscript
(l)
from
the
objects
“Lin
(l)
”,
“Lie
(l)
”,
“New
(l)
”,
“New
tor,(l)
”
introduced
above
and
write:
Lin
U
S
(a/b);
Lin
U
S
/U
S
0
(a/b);
Lie
U
S
(a/b);
Lin
U
S
0
(a/b);
Lie
U
S
/U
S
0
(a/b);
Lie
U
S
0
(a/b)
New
S
∗
(a/b);
New
tor
S
∗
(a/b)
Also,
we
shall
write:
New
Q
S
∗
(a/b)
=
New
S
∗
(a/b)
⊗
Q;
def
Note
that,
for
Z
def
Δ
Lie
U
S
=
Lin
U
S
(1/∞)
×
Lin
US
(1/∞)
Δ
U
S
0
0
b
≥
1,
we
have
a
natural
G
k
-equivariant
inclusion
∼
Lin
U
S
/U
S
0
(b
+
1/∞)
→
Lin
U
S
/U
S
0
(b
+
1/∞)
×
{1}
{1}
→
Lin
U
S
(1/∞)
×
Lin
US
(1/∞)
Δ
U
S
0
0
=
Δ
Lie
U
S
whose
image
forms
a
normal
subgroup
of
Δ
Lie
U
S
;
write
Lie≤b
Δ
Lie
U
S
Δ
U
S
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
57
for
the
quotient
of
Δ
Lie
U
S
by
this
normal
subgroup.
Also,
we
have
a
natural
G
k
-
equivariant
[composite]
inclusion
∼
Lie≤b+1
New
Q
S
∗
(b
+
1/b
+
2)
→
Lie
U
S
/U
S
0
(b
+
1/b
+
2)
→
Lin
U
S
/U
S
0
(b
+
1/b
+
2)
→
Δ
U
S
whose
image
forms
a
normal
subgroup
of
Δ
Lie≤b+1
;
write
U
S
Lie≤b+1
Lie≤b+
Δ
U
Δ
U
S
S
Lie≤b+1
for
the
quotient
of
Δ
U
by
this
normal
subgroup.
Thus,
we
have
natural
G
k
-
S
equivariant
homomorphisms
of
topological
groups:
Lie≤b+
Δ
U
S
→
Δ
Lie
Δ
Lie≤b
Δ
U
S
0
U
S
Δ
U
S
U
S
[the
last
three
of
which
are
easily
verified
to
be
surjective].
Moreover,
forming
the
semi-direct
product
with
G
k
[via
the
natural
actions
of
G
k
]
yields
topological
groups
and
homomorphisms
as
follows:
Lie≤b+
Π
U
S
→
Π
Lie
Π
Lie≤b
Π
U
S
0
U
S
Π
U
S
U
S
Also,
we
note
that
we
have
natural
exact
sequences:
1
→
Lin
U
S
/U
S
0
(1/∞)
→
Δ
Lie
U
S
→
Δ
U
S
0
→
1
1
→
Lin
U
S
/U
S
0
(1/∞)
→
Π
Lie
U
S
→
Π
U
S
0
→
1
Definition
3.5.
Lie≤b
Lie≤b+
Lie
;
Π
Lie≤b
;
Δ
U
;
Π
Lie≤b+
)
(i)
We
shall
refer
to
Δ
Lie
U
S
(respectively,
Π
U
S
;
Δ
U
S
U
S
U
S
S
as
the
[l-adic]
Lie-ification
(respectively,
Lie-ification;
Lie-ification,
truncated
to
order
b;
Lie-ification,
truncated
to
order
b;
Lie-ification,
truncated
to
order
b+;
Lie-
ification,
truncated
to
order
b+)
of
Δ
U
S
(respectively,
Π
U
S
;
Δ
U
S
;
Π
U
S
;
Δ
U
S
;
Π
U
S
)
[over
Δ
U
S
0
(respectively,
Π
U
S
0
;
Δ
U
S
0
;
Π
U
S
0
;
Δ
U
S
0
;
Π
U
S
0
)].
(ii)
Observe
that
it
follows
immediately
from
the
definitions
that,
for
Z
we
have
natural
exact
sequences
b
≥
1,
Lie≤b+1
Lie≤b+
→
Δ
U
→
1
1
→
New
Q
S
∗
(b
+
1/b
+
2)
→
Δ
U
S
S
Lie≤b+1
→
Π
Lie≤b+
→
1
1
→
New
Q
S
∗
(b
+
1/b
+
2)
→
Π
U
S
U
S
on
which
Π
Lie≤b+1
acts
naturally
by
conjugation.
[Here,
we
note
in
passing
that
it
U
S
is
immediate
from
the
definitions
that
the
submodule
New
S
∗
(b
+
1/b
+
2)
⊆
New
Q
S
∗
(b
+
1/b
+
2)
is
contained
in
the
image
of
Δ
U
S
.]
In
particular,
we
obtain
a
natural
inclusion:
Lie≤b+1
New
S
∗
(b
+
1/b
+
2)
→
Δ
U
(⊆
Π
Lie≤b+1
)
U
S
S
58
SHINICHI
MOCHIZUKI
We
shall
refer
to
the
quotients
of
Δ
Lie≤b+1
,
Π
Lie≤b+1
by
the
image
of
this
natural
U
S
U
S
tor≤b+1
tor≤b+1
inclusion
as
the
toral
Lie-ifications
Δ
U
S
,
Π
U
S
of
Δ
U
S
,
Π
U
S
[over
Δ
U
S
0
,
Π
U
S
0
].
Thus,
we
have
natural
exact
sequences
tor≤b+1
Lie≤b+
→
Δ
U
→
1
1
→
New
tor
S
∗
(b
+
1/b
+
2)
→
Δ
U
S
S
tor≤b+1
→
Π
Lie≤b+
→
1
1
→
New
tor
S
∗
(b
+
1/b
+
2)
→
Π
U
S
U
S
acts
naturally
by
conjugation.
on
which
Π
Lie≤b+1
U
S
(iii)
Suppose
that
U
S
→
U
S
0
is
a
connected
finite
étale
covering
that
arises
0
from
an
open
subgroup
Π
U
⊆
Π
U
S
0
;
write
X
→
X
for
the
normalization
of
X
in
S
0
U
S
.
Then
we
shall
say
that
the
[ramified]
covering
X
→
X
is
(S,
S
0
,
Σ)-admissible
0
if
every
closed
point
of
X
that
lies
over
a
point
of
S
is
rational
over
the
base
field
k
of
X
,
and,
moreover,
Π
U
is
a
characteristic
subgroup
of
Π
U
S
0
.
S
0
Remark
3.5.1.
Note
that
it
follows
immediately
from
the
definition
of
Π
Lie
U
S
[cf.
also
Proposition
3.4,
(iii)]
that
we
obtain
a
natural
subgroup
def
D
x
Lie
=
∗
I
x
(l)
[U
S
]
⊗
Q
∗
G
k
⊆
Π
Lie
U
S
which
contains
the
image
of
the
decomposition
group
D
x
∗
[U
S
]
⊆
Π
U
S
via
the
natural
homomorphism
Π
U
S
→
Π
Lie
b
≥
1,
D
x
Lie≤b
⊆
Π
Lie≤b
U
S
.
Let
us
write,
for
Z
U
S
∗
def
def
Lie≤b
Δ
Lie
for
the
image
of
D
x
Lie
in
Π
Lie≤b
;
I
x
Lie
=
D
x
Lie
=
D
x
Lie≤b
U
S
;
I
x
∗
U
S
∗
∗
∗
∗
[Also,
we
shall
use
similar
notation
when
“b”
is
replaced
by
“b+”.]
Proposition
3.6.
Δ
Lie≤b
.
U
S
(Center-freeness
of
Lie-ification)
Δ
Lie
U
S
is
center-free.
Proof.
Since
Δ
U
S
0
is
center-free
[cf.
Proposition
1.8,
(iii)],
and
the
natural
morphism
Δ
Lie
U
S
→
Δ
U
S
0
is
surjective,
it
suffices
to
verify
that
the
centralizer
in
Lie
Lin
U
S
(1/∞)
of
the
image
of
Δ
Lie
U
S
is
trivial.
But
the
image
of
Δ
U
S
in
Lin
U
S
(1/∞)
contains
the
image
of
Δ
U
S
in
Lin
U
S
(1/∞).
In
particular,
it
follows
that
the
cen-
tralizer
in
question
lies
in
the
center
of
Lin
U
S
(1/∞).
Thus,
Proposition
3.6
follows
from
Proposition
3.3,
(ii)
[or,
alternatively,
(iii)].
Remark
3.6.1.
Observe
that
changing
the
choice
of
splitting
G
k
→
D
x
∗
[U
S
]
affects
the
image
of
the
element
F
k
∈
G
k
via
the
composite
of
the
inclusion
G
k
→
Π
U
S
with
the
morphisms
Π
U
S
→
Π
Lie
U
S
;
Π
U
S
→
Π
Lie≤b
;
U
S
Π
U
S
→
Π
Lie≤b+
U
S
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
59
by
conjugation
by
an
element
h
∈
I
x
Lie
,
which,
up
to
a
denominator
dividing
q
k
−
1,
∗
lies
in
the
image
of
I
x
∗
[U
S
]
⊆
Δ
U
S
—
cf.
Proposition
3.4,
(iv);
Proposition
3.6.
In
particular,
it
follows
that
changing
the
choice
of
splittings
G
k
→
D
x
∗
[U
S
]
affects
the
Galois
invariant
splittings
of
Proposition
3.4,
(ii),
by
conjugation
by
h.
Put
another
way,
if
we
identify
the
“Lin
U
S
(1/∞)”,
“Lin
U
S
0
(1/∞)”
portions
of
Δ
Lie
U
S
[cf.
the
definition
of
Δ
Lie
]
with
the
[l-adic
points
of
the
pro-unipotent
algebraic
U
S
groups
determined
by
the]
corresponding
graded
Lie
objects
“Gr
Q
l
(−)(1/∞)”
via
the
Galois
invariant
splittings
of
Proposition
3.4,
(ii),
then
it
follows
that:
Changing
the
choice
of
splitting
G
k
→
D
x
∗
[U
S
]
affects
the
images
of
the
morphisms
Π
U
S
→
Π
Lie
U
S
;
[where
Z
Π
U
S
→
Π
Lie≤b
;
U
S
Π
U
S
→
Π
Lie≤b+
U
S
b
≥
1]
by
conjugation
by
h.
In
light
of
Proposition
3.6,
we
may
apply
the
exact
sequence
“1
→
(−)
→
Aut(−)
→
Out(−)
→
1”
[cf.
§0]
to
construct
the
following
topological
group:
Lie
Δ
LIE
lim
U
S
=
←
−
Aut(Δ
U
)
×
Out(Δ
Lie
)
Gal(X
k
/X
k
)
def
X
U
S
S
[where
X
→
X
ranges
over
the
(S,
S
0
,
Σ)-admissible
coverings
of
X;
U
S
⊆
X
is
the
open
subscheme
determined
by
the
complement
of
the
set
S
of
closed
points
of
X
that
lie
over
points
of
S].
Note
that
G
k
acts
naturally
on
Δ
LIE
U
S
;
thus,
we
may
LIE
form
the
semi-direct
product
of
Δ
U
S
with
G
k
to
obtain
a
topological
group
Π
LIE
U
S
.
Also,
since
the
various
Δ
U
[where
U
S
⊆
X
is
the
open
subscheme
determined
by
S
0
0
the
complement
of
the
set
S
0
of
closed
points
of
X
that
lie
over
points
of
S
0
]
arising
from
the
X
→
X
that
appear
in
this
inverse
limit
are
center-free
[cf.
Proposition
1.8,
(iii)],
the
natural
isomorphism
∼
lim
←−
Aut(Δ
U
S
0
)
×
Out(Δ
U
S
)
Gal(X
k
/X
k
)
→
Δ
U
S
0
X
0
LIE
determines
surjections
Δ
LIE
U
S
Δ
U
S
0
,
Π
U
S
Π
U
S
0
.
Next,
let
us
observe
that,
for
Z
b
≥
1,
the
various
quotients
Δ
Lie
U
S
Δ
tor≤b+1
Δ
Lie≤b+
Δ
Lie≤b
determine
quotients
of
topological
groups
Δ
LIE
U
S
U
U
U
S
S
S
LIE≤b+
TOR≤b+1
Δ
U
Δ
LIE≤b
,
Π
LIE
Π
LIE≤b+
Π
LIE≤b
.
Δ
TOR≤b+1
U
S
Π
U
S
U
S
U
S
U
S
U
S
S
Thus,
we
obtain
natural
homomorphisms
of
topological
groups:
TOR≤b+1
LIE≤b+
Δ
U
Δ
LIE≤b
Δ
U
S
0
Δ
U
S
→
Δ
LIE
U
S
Δ
U
S
U
S
S
TOR≤b+1
LIE≤b+
Π
U
S
→
Π
LIE
Π
U
Π
LIE≤b
Π
U
S
0
U
S
Π
U
S
U
S
S
We
shall
denote
by
LIE≤b+
Δ
≤b+
;
U
S
⊆
Δ
U
S
LIE≤b+
Π
≤b+
;
U
S
⊆
Π
U
S
LIE≤b
Δ
≤b
;
U
S
⊆
Δ
U
S
LIE≤b
Π
≤b
U
S
⊆
Π
U
S
60
SHINICHI
MOCHIZUKI
the
respective
images
of
Δ
U
S
,
Π
U
S
via
these
natural
homomorphisms.
Thus,
one
≤b
may
think
of
Δ
≤b
U
S
,
Π
U
S
as
being
a
sort
of
“canonical
integral
structure”
on
the
“inverse
limit
truncated
Lie-ifications”
Δ
LIE≤b
,
Π
LIE≤b
.
U
S
U
S
Here,
we
note
in
passing,
relative
to
the
theory
of
§1,
2,
that
[it
is
immediate
from
the
definitions
that]
when
S
=
S
∗
[so
U
S
0
=
X],
the
quotient
Π
U
S
Π
≤2
U
S
is
the
maximal
cuspidally
pro-l
abelian
quotient
of
Π
U
S
[cf.
Proposition
1.14,
(i)].
LIE
Next,
let
us
observe
that
in
the
inverse
limit
used
to
define
Δ
LIE
U
S
,
Π
U
S
,
the
various
“I
x
Lie
”,
“D
x
Lie
”
[cf.
Remark
3.5.1]
form
a
compatible
system,
hence
give
rise
∗
∗
to
subgroups
⊆
D
x
LIE
⊆
Π
LIE
I
x
LIE
U
S
;
∗
∗
I
x
LIE≤b
⊆
D
x
LIE≤b
⊆
Π
LIE≤b
U
S
∗
∗
together
with
natural
exact
sequences
and
isomorphisms
[when
b
≥
2]
→
D
x
LIE
→
G
k
→
1
1
→
I
x
LIE
∗
∗
→
D
x
LIE≤b
→
G
k
→
1
1
→
I
x
LIE≤b
∗
∗
∼
∼
I
x
LIE
=
I
x
LIE≤b
=
I
x
∗
[U
S
]
⊗
Q
∗
∗
(l)
[and
similarly
when
“b”
is
replaced
by
“b+”].
Also,
the
images
of
the
subgroups
I
x
∗
[U
S
],
D
x
∗
[U
S
]
of
Π
U
S
determine
subgroups
⊆
D
x
≤b
⊆
Π
≤b
I
x
≤b
U
S
∗
∗
[and
similarly
when
“b”
is
replaced
by
“b+”].
In
the
following,
let
us
write
[cf.
Proposition
3.3,
(iv)]
def
(l)
Cnw
S
∗
(a/b)
=
Cnw
S
∗
(a/b);
Cnw
Q
S
∗
(a/b)
=
Cnw
S
∗
(a/b)
⊗
Q
def
(l)
[where
a,
b
∈
Z,
1
≤
a
≤
b].
Before
proceeding,
let
us
observe
that
[it
is
immediate
from
the
definitions
that]
the
natural
surjections
Δ
LIE≤1
Δ
U
S
0
;
Δ
LIE≤1+
U
S
U
S
Π
LIE≤1+
Π
LIE≤1
Π
U
S
0
U
S
U
S
are
isomorphisms.
On
the
other
hand,
for
b
≥
2,
we
have
the
following
result:
Proposition
3.7.
Z
b
≥
2:
(Plus
Liftings
of
Canonical
Integral
Structures)
For
≤b
≤b+
≤b
(i)
The
natural
surjections
Δ
≤b+
U
S
Δ
U
S
,
Π
U
S
Π
U
S
are
isomorphisms.
LIE≤b
(ii)
Any
two
liftings
of
the
natural
inclusion
Π
≤b
to
inclusions
U
S
→
Π
U
S
≤b
LIE≤b+
LIE≤b+
Π
U
S
→
Π
U
S
differ
by
conjugation
in
Π
U
S
by
a
unique
element
of
the
LIE≤b+
LIE≤b
kernel
of
Π
U
S
Π
U
S
.
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
61
LIE≤b
(iii)
Any
two
liftings
of
the
natural
inclusion
Π
≤b
to
inclusions
U
S
→
Π
U
S
≤b
LIE≤b+
≤b+
Π
U
S
→
Π
U
S
whose
images
contain
D
x
∗
in
fact
coincide.
Proof.
First,
we
consider
assertion
(i).
It
follows
immediately
from
the
definitions
that
the
kernel
in
question
≤b
≤b+
≤b
Ker(Δ
≤b+
U
S
Δ
U
S
)
=
Ker(Π
U
S
Π
U
S
)
is
contained
in
[and,
in
fact,
equal
to]
the
inverse
limit
lim
←−
Cnw
S
∗
(b
+
1/b
+
2)
X
[where
X
→
X
ranges
over
the
(S,
S
0
,
Σ)-admissible
coverings
of
X;
S
∗
(respec-
tively,
S
)
is
the
set
of
closed
points
of
X
that
lie
over
points
of
S
∗
(respectively,
S)].
On
the
other
hand,
it
follows
from
the
definition
of
“Cnw
S
∗
(b
+
1/b
+
2)”
that
Cnw
S
∗
(b
+
1/b
+
2)
is
generated
by
certain
successive
brackets
of
the
var-
(l)
ious
generators
of
the
Lie
algebra
Gr(Δ
U
)
[cf.
Proposition
3.3,
(i)]
with
the
S
property
that
at
least
one
of
the
generators
appearing
in
the
successive
bracket
is
[in
the
notation
of
Proposition
3.3,
(i)]
either
one
of
the
[analogue
for
X
of
the]
def
“α
1
,
.
.
.
,
α
g
,
β
1
,
.
.
.
,
β
g
”
or
one
of
the
“ζ
x
”,
where
x
∈
S
0
=
S
\S
∗
.
Moreover,
since,
by
taking
Π
U
⊆
Π
U
to
be
sufficiently
small,
one
may
arrange
that
the
im-
S
S
0
0
(l)
(l)
age
of
Δ
U
(1/3)
in
Δ
U
(1/3)
be
contained
in
an
arbitrarily
small
open
subgroup
S
S
0
0
(l)
of
Δ
U
(1/3),
it
thus
follows
that
the
above
inverse
limit
vanishes.
This
completes
S
0
the
proof
of
assertion
(i).
Next,
let
us
observe
that
to
prove
assertion
(ii),
it
suffices
—
in
light
of
the
natural
isomorphism
∼
LIE≤b+
Q
Π
LIE≤b
)
→
lim
Ker(Π
U
U
S
←−
Cnw
S
(b
+
1/b
+
2)
S
X
∗
[where
X
,
S
∗
are
as
above]
—
to
show
that
Q
H
i
(Π
≤b
U
S
,
Cnw
S
∗
(b
+
1/b
+
2))
=
0
Q
for
i
=
0,
1,
each
S
∗
as
above.
Since
the
action
of
Δ
≤b
U
S
on
Cnw
S
(b
+
1/b
+
2)
∗
clearly
factors
through
a
finite
quotient
of
Δ
≤b
U
S
Δ
U
S
0
,
it
thus
suffices
to
observe
[by
considering
the
Leray
spectral
sequence
associated
to
the
surjection
Π
≤b
U
S
G
k
]
Q
that
the
action
of
F
k
on
Cnw
S
∗
(b+1/b+2)
is
“of
weight
b+1
≥
3”,
while
the
action
(l)
of
F
k
on
(Δ
U
)
ab
is
“of
weight
≤
2”
[cf.
Proposition
3.4,
(i)].
This
completes
the
S
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
First,
let
us
observe
that
any
two
liftings
of
LIE≤b
LIE≤b+
the
natural
inclusion
Π
≤b
to
inclusions
Π
≤b
whose
images
U
S
→
Π
U
S
U
S
→
Π
U
S
62
SHINICHI
MOCHIZUKI
∼
contain
D
x
≤b+
→
D
x
≤b
[since
b
≥
2]
in
fact
coincide
on
D
x
≤b
⊆
Π
≤b
U
S
.
Thus,
by
∗
∗
∗
assertion
(ii),
it
suffices
to
verify
that
the
submodule
of
F
k
-invariants
of
LIE≤b+
Π
LIE≤b
)
Ker(Π
U
U
S
S
is
zero.
But
in
light
of
the
natural
isomorphism
∼
LIE≤b+
Q
Π
LIE≤b
)
→
lim
Ker(Π
U
U
S
←−
Cnw
S
(b
+
1/b
+
2)
S
∗
X
[where
X
,
S
∗
are
as
above],
this
follows
from
Proposition
3.4,
(i).
This
completes
the
proof
of
assertion
(iii).
Next,
for
Z
b
≥
1,
let
us
denote
by
⊆
Δ
TOR≤b+1
;
Δ
≤b++
U
S
U
S
Π
≤b++
⊆
Π
TOR≤b+1
U
S
U
S
the
respective
images
of
Δ
U
S
,
Π
U
S
via
the
natural
homomorphisms
considered
above
and
by
⊆
D
x
≤b++
⊆
Π
≤b++
I
x
≤b++
U
S
∗
∗
the
images
of
the
subgroups
I
x
∗
[U
S
],
D
x
∗
[U
S
]
of
Π
U
S
.
Observe
that
it
follows
from
,
Π
TOR≤b+1
[cf.
also
Proposition
3.3,
(iv)]
that
the
the
definition
of
Δ
TOR≤b+1
U
S
U
S
≤b++
≤b+
≤b++
natural
surjections
Δ
U
S
Δ
U
S
,
Π
U
S
Π
≤b+
are,
in
fact,
isomorphisms.
U
S
Thus,
by
Proposition
3.7,
(i),
we
obtain
a
commutative
diagram
of
natural
homo-
morphisms
Π
≤b+1
U
S
⏐
⏐
LIE≤b+1
Π
U
S
Π
≤b++
U
S
⏐
⏐
Π
TOR≤b+1
U
S
→
∼
Π
≤b+
U
S
⏐
⏐
→
∼
Π
≤b
U
⏐
S
⏐
Π
LIE≤b+
U
S
Π
LIE≤b
U
S
[where
the
vertical
arrows
are
the
natural
inclusions;
all
of
the
horizontal
arrows
are
surjections;
the
second
two
upper
horizontal
arrows
are
isomorphisms].
Moreover,
it
follows
immediately
from
the
definitions
that
the
first
square
in
this
commuta-
⊆
Π
LIE≤b+1
may
be
tive
diagram
is
cartesian.
That
is
to
say,
the
subgroup
Π
≤b+1
U
S
U
S
LIE≤b+1
TOR≤b+1
thought
of
as
the
inverse
image
via
the
natural
surjection
Π
U
S
Π
U
S
≤b
LIE≤b+
of
the
image
of
a
certain
lifting
of
the
natural
inclusion
Π
U
S
→
Π
U
S
[cf.
Propo-
≤b
TOR≤b+1
sition
3.7,
(i)]
to
an
inclusion
Π
U
S
→
Π
U
S
.
Also,
let
us
write:
≤b
Π
≤b
U
S
[csp]
=
Ker(Π
U
S
Π
X
)
def
[csp]
=
Ker(Π
≤b++
Π
X
)
Π
≤b++
U
S
U
S
def
≤b++
for
the
cuspidal
subgroups
of
Π
≤b
.
U
S
,
Π
U
S
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
63
Next,
following
the
pattern
of
§1,
we
relate
the
constructions
made
so
far
to
the
fundamental
groups
Δ
U
X×X
,
Π
U
X×X
[cf.
the
discussion
preceding
Proposition
1.6].
For
simplicity,
we
assume
from
now
on
that:
S
=
S
∗
=
{x
∗
}
[so
S
0
=
∅].
Write
D
x
∗
[X]
⊆
Π
X
for
the
image
of
D
x
∗
[U
S
]
via
the
natural
surjection
Π
U
S
Π
X
.
Then
the
projection
Π
U
X×X
Π
X
to
the
second
factor
determines
a
natural
isomorphism
∼
Π
U
S
→
Π
U
X×X
×
Π
X
D
x
∗
[X]
[cf.
Proposition
1.8,
(ii)].
Moreover,
this
isomorphism
determines
a
natural
iso-
morphism
∼
(Π
U
S
⊇)
D
x
∗
[U
S
]
→
D
X
×
Π
X
D
x
∗
[X]
(⊆
D
X
⊆
Π
U
X×X
)
[where
“D
X
”
is
as
in
the
discussion
preceding
Proposition
1.12]
which
is
compat-
ible
with
the
natural
inclusions
D
x
∗
[U
S
]
→
Π
U
S
,
D
X
→
Π
U
X×X
.
Put
another
way,
D
x
∗
[U
S
]
[hence
also
I
x
∗
[U
S
],
G
k
⊆
D
x
∗
[U
S
]]
may
be
thought
of
as
being
“si-
multaneously”
a
subgroup
of
both
Π
U
S
and
D
X
.
Thus,
we
obtain
a
natural
exact
sequence
1
−→
Δ
U
S
−→
Π
U
X×X
−→
Π
X
−→
1
together
with
compatible
inclusions
Δ
U
S
⊇
Δ
U
⊇
I
x
∗
[U
S
]
⊆
D
x
∗
[U
S
]
⊆
D
X
⊆
Π
U
X×X
S
[where
X
→
X
is
an
(S,
∅,
Σ)-admissible
covering
of
X;
U
S
⊆
X
is
the
open
subscheme
determined
by
the
complement
of
the
set
S
of
closed
points
of
X
that
lie
over
x
∗
].
Also,
we
shall
write:
def
Δ
D
X
=
D
X
Δ
U
X×X
⊆
Π
U
X×X
In
particular,
we
obtain
natural
actions
[by
conjugation]
of
D
X
on
Δ
U
S
,
Δ
U
S
[as
well
as
on
the
various
objects
naturally
constructed
from
Δ
U
S
,
Δ
U
in
the
above
S
discussion],
which
we
shall
refer
to
as
diagonal
actions.
Proposition
3.8.
(Characterization
of
the
Diagonal
Action)
Suppose
that
S
=
S
∗
=
{x
∗
}.
Then
in
the
notation
and
terminology
of
the
above
discussion,
the
diagonal
action
of
D
X
on
Lin
U
S
(1/∞)
is
completely
determined
[i.e.,
as
a
continuous
action
of
the
topological
group
D
X
on
the
topological
group
Lin
U
S
(1/∞)]
by
the
following
conditions:
(a)
the
action
is
compatible
with
the
natural
action
of
D
x
∗
[U
S
]
⊆
D
X
on
Lin
U
S
(1/∞);
(b)
the
action
is
compatible
with
the
filtration
{Lin
U
S
(a/∞)}
[where
a
≥
1
is
an
integer]
on
Lin
U
S
(1/∞).
64
SHINICHI
MOCHIZUKI
(c)
the
action
coincides
with
the
diagonal
action
of
D
X
on
the
quotient
Lin
U
S
(1/4)
[cf.
condition
(b)]
of
Lin
U
S
(1/∞).
Proof.
First,
I
claim
that
it
suffices
to
show
that
these
conditions
determine
the
def
Δ
Δ
Δ
action
of
the
subgroup
D
X
/X
=
D
X
×
Δ
X
Δ
X
⊆
D
X
⊆
D
X
on
Lin
U
(1/∞).
S
Δ
is
determined,
it
follows
that
the
action
of
Indeed,
once
the
action
of
D
X
/X
def
Δ
D
X
/X
=
D
X
×
Π
X
Π
X
⊆
D
x
∗
[U
S
]
·
D
X
/X
⊆
D
X
is
determined
[cf.
condition
(a)].
On
the
other
hand,
since
Π
X
is
an
open
normal
subgroup
of
Π
X
,
it
follows
that
D
X
/X
is
an
open
normal
subgroup
of
D
X
.
Thus,
by
considering
the
conjugation
actions
of
D
X
on
D
X
/X
and
of
Im(D
X
)
⊆
Lin
U
(1/∞)
S
on
Im(D
X
/X
)
⊆
Lin
U
(1/∞)
[i.e.,
of
the
group
of
automorphisms
of
Lin
U
(1/∞)
S
S
induced
by
elements
of
D
X
on
the
group
of
automorphisms
of
Lin
U
(1/∞)
induced
S
by
elements
of
D
X
/X
],
we
conclude
that
the
action
of
D
X
on
Lin
U
S
(1/∞)
is
de-
termined
up
to
composition
with
automorphisms
of
Lin
U
S
(1/∞)
that
commute
with
the
action
of
D
X
/X
and
[cf.
condition
(c)]
induce
the
identity
on
the
quo-
tient
Lin
U
S
(1/4).
Now
let
α
be
an
automorphism
of
Lin
U
S
(1/∞)
that
commutes
with
the
action
of
D
X
/X
and
induces
the
identity
on
the
quotient
Lin
U
S
(1/4).
Then
α
commutes
with
some
open
subgroup
of
G
k
⊆
D
x
∗
[U
S
]
⊆
D
X
,
so
α
induces
an
automorphism
of
Lie
U
S
(1/∞)
that
is
compatible
with
the
splittings
of
Propo-
(l)
sition
3.4,
(ii).
Since
Gr(Δ
U
)
is
generated
by
its
elements
“of
weight
≤
2”
[cf.
S
Proposition
3.3,
(i)],
we
thus
conclude
that
α
induces
the
identity
automorphism
of
Lie
U
(1/∞),
hence
that
α
itself
is
the
identity
automorphism.
This
completes
S
the
proof
of
the
claim.
Next,
let
us
observe
that
by
condition
(c)
[cf.
also
Proposition
3.3,
(i)],
the
Δ
action
of
D
X
/X
on
Lin
U
(1/∞)
is
unipotent,
relative
to
the
filtration
of
condition
S
(b).
Thus,
it
follows
[from
the
definition
of
“Lie(−)”]
that
the
induced
action
of
Δ
D
X
/X
on
Lie
U
(1/∞)
determines
an
action
of
the
Lie
algebra
S
def
Δ
Δ
(l)
(1/∞))
Lie(D
X
/X
)
=
Lie((D
X
/X
)
Δ
(l)
Δ
(l)
for
the
maximal
pro-l
quotient
of
(D
X
]
on
the
Lie
[where
we
write
(D
X
/X
)
/X
)
algebra
Lie
U
(1/∞).
Moreover,
to
complete
the
proof
of
Proposition
3.8,
it
suffices
S
to
show
that
this
Lie
algebra
action
is
the
action
arising
from
the
diagonal
action.
In
fact,
since
this
Lie
algebra
action
is
compatible
[cf.
condition
(a)]
with
the
actions
of
Δ
G
k
on
Lie(D
X
/X
),
Lie
U
(1/∞),
it
follows,
by
considering
the
induced
eigenspace
S
splittings
[cf.
Proposition
3.4,
(ii)],
that
[to
complete
the
proof
of
Proposition
3.8]
def
Δ
Δ
it
suffices
to
show
that
the
Lie
algebra
action
of
Gr(D
X
/X
)
=
Gr(Lie(D
X
/X
))
on
(l)
Gr(Δ
U
)
is
the
action
arising
from
the
diagonal
action.
On
the
other
hand,
since
S
(l)
Δ
Gr(D
X
/X
),
Gr(Δ
U
)
are
generated
by
elements
“of
weight
≤
2”
[cf.
S
Proposition
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
65
3.3,
(i)],
this
follows
by
observing
that
the
Lie
algebra
action
of
the
unique
generator
Δ
Δ
of
Gr(D
X
/X
)
“of
weight
2”
[which
arises
from
I
x
∗
[U
S
]
⊆
D
X
/X
]
is
determined
by
Δ
condition
(a),
while
the
Lie
algebra
action
of
the
generators
of
Gr(D
X
/X
)
“of
weight
(l)
(l)
S
S
1”
[which
send
elements
of
Gr(Δ
U
)
“of
weight
≤
2”
to
elements
of
Gr(Δ
U
)
“of
weight
≤
3”]
is
determined
by
condition
(c).
This
completes
the
proof
of
Proposition
3.8.
Remark
3.8.1.
Note
that
the
conditions
of
Proposition
3.8
allow
one
to
char-
acterize
not
only
the
diagonal
action
of
D
X
on
Lin
U
S
(1/∞),
but
also
on
Δ
Lie
U
,
S
LIE
Π
Lie
,
Π
LIE
,
hence
also
on
Δ
U
U
S
[where
we
note
that
the
diagonal
action
of
D
X
on
U
S
S
Gal(X
k
/X
k
)
is
simply
the
conjugation
action
arising
from
the
quotients
D
X
Π
X
,
Δ
X
Gal(X
k
/X
k
)].
Remark
3.8.2.
Note
that
the
groups
Lin
U
S
(1/4)
of
condition
(c)
of
Proposition
3.8
are,
as
groups
equipped
with
the
surjection
Lin
U
S
(1/4)
Lin
X
(1/4),
cuspi-
dally
abelian
[i.e.,
the
kernel
of
this
surjection
is
abelian],
hence
may
be
constructed
from
the
maximal
cuspidally
abelian
quotients
Π
U
X×X
Π
c-ab
U
X×X
of
Theorem
1.16.
Proposition
3.9.
(Extensions
of
Canonical
Integral
Structures)
Suppose
that
S
=
S
∗
=
{x
∗
}
[cf.
Remark
3.9.2
below].
Let
b
≥
1
be
an
integer.
Then:
(i)
Suppose
that
b
=
1.
Then
any
two
liftings
of
the
natural
inclusion
Π
≤b
U
S
→
LIE≤b+
≤b
TOR≤b+1
TOR≤b+1
Π
U
S
to
inclusions
Π
U
S
→
Π
U
S
differ
by
conjugation
in
Π
U
S
by
TOR≤b+1
LIE≤b+
an
element
of
the
kernel
of
Π
U
S
Π
U
S
.
(ii)
Suppose
that
b
≥
2.
Then
any
two
liftings
of
the
natural
inclusion
Π
≤b
U
S
→
LIE≤b+
≤b
TOR≤b+1
Π
U
S
to
inclusions
Π
U
S
→
Π
U
S
whose
images
contain
I
x
≤b++
differ
by
∗
TOR≤b+1
TOR≤b+1
LIE≤b+
conjugation
in
Π
U
S
by
an
element
of
the
kernel
of
Π
U
S
Π
U
.
S
that
satisfies
the
(iii)
Let
β
be
an
automorphism
of
the
profinite
group
Π
≤b+1
U
S
following
two
conditions:
(a)
β
preserves
and
induces
the
identity
on
the
quo-
≤b+1
tient
Π
≤b+1
Π
≤b
⊆
Π
≤b+1
.
Then
β
is
a
U
S
U
S
;
(b)
β
preserves
the
subgroup
I
x
∗
U
S
≤b+1
≤b
Ker(Π
U
S
Π
U
S
)-inner
automorphism.
LIE≤b+
Π
U
)
be
an
element
that
is
invariant
(iv)
Let
δ
∈
Ker(Π
TOR≤b+1
U
S
S
under
the
diagonal
action
of
D
X
.
Then
if
b
=
1,
then
δ
lies
in
the
image
of
I
x
∗
[U
S
]
⊗
(Q
l
/Z
l
);
if
b
≥
2,
then
δ
is
the
identity
element.
(v)
Write
≤b
lim
Π
U
S
Π
≤∞
U
S
=
←
−
Π
U
S
;
def
b
≤b
Δ
U
S
Δ
≤∞
lim
U
S
=
←
−
Δ
U
S
def
b
66
SHINICHI
MOCHIZUKI
≤b
for
the
the
quotients
of
Π
U
S
,
Δ
U
S
defined
by
the
inverse
limit
of
the
Π
≤b
U
S
,
Δ
U
S
and
Π
U
X×X
Π
≤∞
U
X×X
;
Δ
U
X×X
Δ
≤∞
U
X×X
for
the
quotients
of
Π
U
X×X
,
Δ
U
X×X
determined
by
the
kernel
in
Δ
U
S
⊆
Δ
U
X×X
⊆
Π
U
X×X
[cf.
the
discussion
preceding
Proposition
3.8]
of
Ker(Δ
U
S
Δ
≤∞
U
S
).
Then
≤∞
≤∞
≤∞
Π
U
S
Π
U
S
(respectively,
Δ
U
S
Δ
U
S
;
Π
U
X×X
Π
U
X×X
;
Δ
U
X×X
Δ
≤∞
U
X×X
)
is
the
maximal
cuspidally
pro-l
quotient
of
Π
U
S
(respectively,
Δ
U
S
;
Π
U
X×X
;
≤∞
≤∞
≤∞
†
†
Δ
U
X×X
);
moreover,
(Π
≤∞
U
S
)
,
Δ
U
S
,
(Π
U
X×X
)
,
Δ
U
X×X
[where
the
daggers
denote
the
result
of
applying
the
operation
“×
G
k
G
†
k
”]
are
slim.
Proof.
First,
we
consider
assertions
(i),
(ii).
Observe
that,
for
Z
b
≥
1,
the
≤b
LIE≤b+
difference
of
any
two
liftings
of
the
natural
inclusion
Π
U
S
→
Π
U
S
to
inclusions
≤b
TOR≤b+1
Π
U
S
→
Π
U
S
determines
a
compatible
collection
of
cohomology
classes
tor
η
S
∈
H
1
(Π
≤b
U
S
,
New
S
(b
+
1/b
+
2))
[where
X
→
X
ranges
over
the
(S,
∅,
Σ)-admissible
coverings
of
X;
S
is
the
set
of
closed
points
of
X
that
lie
over
x
∗
].
Since
New
tor
S
(b
+
1/b
+
2)
=
0
whenever
b
is
even,
we
may
assume
for
the
remainder
of
the
proof
of
assertions
(i),
(ii)
that
b
is
odd.
Next,
let
us
observe
that
by
Proposition
3.4,
(i),
the
zeroth
cohomology
module
tor
H
0
(Π
≤b
U
S
,
New
S
(b
+
1/b
+
2))
is
finite.
This
finiteness
implies
that
any
[not
necessarily
compatible!]
system
of
tor
sections
of
a
compatible
system
of
torsors
over
H
0
(Π
≤b
U
S
,
New
S
(b
+
1/b
+
2))
always
admits
a
compatible
cofinal
subsystem.
In
light
of
the
natural
isomorphism
∼
LIE≤b+
tor
Π
U
)
→
lim
Ker(Π
TOR≤b+1
U
S
←−
New
S
(b
+
1/b
+
2))
S
X
[where
X
,
S
are
as
described
above],
we
thus
conclude
that
in
order
to
show
TOR≤b+1
that
the
two
inclusions
Π
≤b
differ
by
conjugation
by
an
element
of
U
S
→
Π
U
S
TOR≤b+1
LIE≤b+
Ker(Π
U
S
Π
U
S
),
it
suffices
to
show
that
the
η
S
=
0.
tor
Note
that
Π
≤b
U
S
[csp]
acts
trivially
on
New
S
(b
+
1/b
+
2)).
Now
I
claim
that:
Each
η
S
arises
from
a
unique
class
[which,
by
abuse
of
notation,
we
shall
also
denote
by
η
S
]
in
H
1
(Π
X
,
New
tor
S
(b
+
1/b
+
2)).
Indeed,
if
b
=
1,
this
claim
follows
from
the
fact
that
Π
≤b
U
S
[csp]
=
{1}
[cf.
the
discus-
sion
preceding
Proposition
3.7],
so
assume
that
b
≥
2,
and
that
we
are
in
the
situa-
tion
of
assertion
(ii).
Now
observe
that
since
S
=
S
∗
is
of
cardinality
one,
it
follows
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
67
≤b++
that
Π
≤b
[csp])
is
topologically
generated
by
the
Π
≤b
U
S
[csp]
(respectively,
Π
U
S
U
S
-
(re-
≤b++
≤b
≤b++
spectively,
Π
U
S
-)
conjugates
of
I
x
∗
(respectively,
I
x
∗
).
Note,
moreover,
that
it
LIE≤b+
is
immediate
from
the
definitions
that
every
element
of
Ker(Π
TOR≤b+1
Π
U
)
U
S
S
≤b
≤b++
.
In
particular,
it
follows
that
the
images
of
Π
U
S
[csp]
via
the
commutes
with
I
x
∗
≤b
two
inclusions
Π
U
S
→
Π
TOR≤b+1
under
consideration
necessarily
coincide.
But
this
U
S
implies
that
each
η
S
arises
from
a
unique
class
in
H
1
(Π
X
,
New
tor
S
(b
+
1/b
+
2)),
thus
completing
the
proof
of
the
claim.
Next,
[returning
to
the
general
situation
involving
both
assertions
(i)
and
(ii)]
let
X
→
X
be
a
morphism
of
(S,
∅,
Σ)-admissible
coverings
of
X.
Write
U
S
⊆
X
for
the
open
subscheme
determined
by
the
complement
of
the
set
S
of
closed
points
of
X
that
lie
over
points
of
S.
Also,
let
us
assume
that
the
open
subgroup
Δ
X
⊆
Δ
X
arises
from
some
open
subgroup
H
⊆
Δ
ab
X
that
is
preserved
by
the
action
of
Π
X
.
Thus,
it
def
follows
that
the
covering
X
k
→
X
k
is
abelian;
write
Gal(X
/X
)
=
Gal(X
k
/X
k
).
For
c
a
positive
integer,
set:
def
R
=
Z
l
;
def
def
R
c
=
Z
l
[c
·
Gal(X
/X
)]
⊆
R
=
Z
l
[Gal(X
/X
)]
[where
we
write
c·Gal(X
/X
)
⊆
Gal(X
/X
)
for
the
subgroup
of
the
abelian
group
Gal(X
/X
)
that
arises
as
the
image
of
multiplication
by
c].
Thus,
R
(respectively,
R
;
R
c
)
is
a
commutative
ring
with
unity
whose
underlying
R
-
(respectively,
R
c
-;
R
-)
module
is
finite
and
free;
moreover,
R
,
R
c
admit
a
natural
Π
X
-action
[induced
by
the
conjugation
action
of
Π
X
on
the
subquotient
Gal(X
/X
)
of
Π
X
].
Also,
we
shall
denote
by
c
:
R
c
R
;
:
R
R
the
augmentations
obtained
by
mapping
all
of
the
elements
of
Gal(X
/X
)
to
1.
Next,
let
us
observe
that
S
,
S
admit
natural
Π
X
-actions
with
respect
to
which
we
have
natural
isomorphisms
of
Π
X
-modules
[cf.
Proposition
3.3,
(i),
(iv)]
∼
New
S
(2/3)
→
R
[S
]
⊗
M
X
;
(l)
∼
New
S
(2/3)
→
R
[S
]
⊗
M
X
(l)
which
determine
natural
isomorphisms
of
Π
X
-modules
∼
(l)
∼
(l)
New
S
(2c/2c
+
1)
→
Lie
cR
(R
[S
]
⊗
M
X
)
New
S
(2c/2c
+
1)
→
Lie
cR
(R
[S
]
⊗
M
X
)
[cf.
the
notation
of
Proposition
A.1]
for
integers
c
≥
1.
In
the
following,
we
shall
identify
the
domains
and
codomains
of
these
isomorphisms
via
these
isomorphisms.
Next,
let
us
observe
that
the
R
-module
R
[S
]
admits
a
natural
R
-module
structure
that
is
compatible
with
the
Π
X
-action
on
R
,
R
[S
].
Note,
moreover,
that
R
[S
]
is
a
free
R
-module,
and
that
we
have
a
natural
isomorphism
∼
R
[S
]
⊗
R
,
R
→
R
[S
]
68
SHINICHI
MOCHIZUKI
induced
by
the
augmentation
:
R
R
.
Also,
we
observe
that
any
choice
of
rep-
resentatives
in
S
of
the
Δ
X
/Δ
X
=
Gal(X
/X
)-orbits
of
S
[where
we
note
that
the
set
of
such
orbits
may
be
naturally
identified
with
S
]
determines
an
R
-basis
of
R
[S
],
hence
[by
considering
“Hall
bases”
—
cf.,
e.g.,
[Bour],
Chapter
II,
§2.11]
an
R
-basis
of
Lie
cR
(R
[S
]).
Note
that
since
the
natural
action
of
Gal(X
/X
)
on
Lie
cR
(R
[S
])
is
compatible
with
the
Lie
algebra
structure,
it
follows
that:
This
natural
action
of
Gal(X
/X
)
on
Lie
cR
(R
[S
])
is
given
by
composing
the
R
-module
structure
action
Gal(X
/X
)
→
R
with
the
morphism
c·
:
Gal(X
/X
)
→
Gal(X
/X
)
given
by
multiplication
by
c.
In
particular,
this
natural
action
of
Gal(X
/X
)
on
Lie
cR
(R
[S
])
factors
through
the
quotient
Gal(X
/X
)
c·Gal(X
/X
)
and
hence
determines
on
Lie
cR
(R
[S
])
a
structure
of
“induced”
c·
Gal(X
/X
)-module
[in
the
terminology
of
the
cohomol-
ogy
theory
of
finite
groups].
Thus,
we
obtain
natural,
Π
X
-equivariant
isomorphisms
∼
R
[S
]
→
R
[S
]
Gal(X
/X
)
∼
Lie
cR
(R
[S
])
⊗
R
c
,
c
R
→
Lie
cR
(R
[S
])
Gal(X
/X
)
=
Lie
cR
(R
[S
])
c·Gal(X
/X
)
[where
we
use
superscripts
to
denote
the
submodules
of
invariants
with
respect
to
the
action
of
the
superscripted
group].
Moreover,
we
observe
that
relative
to
these
natural
isomorphisms,
the
restriction
of
the
natural
surjection
Lie
cR
(R
[S
])
Lie
cR
(R
[S
])
⊗
R
c
,
c
R
to
the
submodule
of
Gal(X
/X
)-invariants
induces
the
endomorphism
of
the
module
Lie
cR
(R
[S
])
⊗
R
c
,
c
R
given
by
multiplication
by
the
order
of
c
·
Gal(X
/X
).
Now
let
us
write:
def
c
New
tor
S
/S
(2c/2c
+
1)
=
Lie
R
(R
[S
]
⊗
M
X
)
⊗
(Q
l
/Z
l
)
(l)
def
tor
R
New
tor
S
/S
(2c/2c
+
1)
=
New
S
/S
(2c/2c
+
1)
⊗
R
c
,
c
[where
c
≥
1
is
an
integer].
Then
in
light
of
the
above
observations
[together
with
Propositions
A.1,
(iv);
3.3,
(iv)],
we
conclude
the
following:
(A)
The
natural
surjection
of
Π
X
-modules
tor
New
tor
S
(b
+
1/b
+
2)
New
S
(b
+
1/b
+
2)
admits
a
factorization
tor
tor
New
tor
S
(b
+
1/b
+
2)
New
S
/S
(b
+
1/b
+
2)
New
S
/S
(b
+
1/b
+
2)
New
tor
S
(b
+
1/b
+
2)
[via
morphisms
of
Π
X
-modules].
Moreover,
the
natural
action
of
Δ
X
on
the
module
New
tor
S
/S
(b
+
1/b
+
2)
factors
through
the
quotient
Δ
X
Gal(X
/X
)
c
·
Gal(X
/X
)
and
determines
on
New
tor
S
/S
(b
+
1/b
+
2)
a
structure
of
induced
c
·
Gal(X
/X
)-module.
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
69
(B)
The
induced
morphism
on
Δ
X
-invariants
Δ
X
Δ
X
New
tor
→
New
tor
=
New
tor
S
(b
+
1/b
+
2)
S
(b
+
1/b
+
2)
S
(b
+
1/b
+
2)
of
the
[first]
natural
surjection
of
(A)
factors,
in
a
Π
X
-equivariant
fashion,
through
the
endomorphism
tor
New
tor
S
/S
(b
+
1/b
+
2)
→
New
S
/S
(b
+
1/b
+
2)
tor
[hence
also
through
the
endomorphism
New
tor
S
(b
+
1/b
+
2)
→
New
S
(b
+
1/b
+
2)]
given
by
multiplication
by
the
order
of
c
·
Gal(X
/X
).
Also,
before
proceeding,
we
make
the
following
elementary
observation
con-
cerning
the
group
cohomology
of
induced
modules:
ab
(C)
Suppose
that
H
=
l
n
·
Δ
ab
X
⊆
Δ
X
,
where
n
is
a
positive
integer.
For
M
a
finitely
generated
Z
l
-module
[which
we
regard
as
equipped
with
the
trivial
Δ
X
-action],
write:
def
(l)
H
X
=
H
1
(Δ
X
,
M
⊗
M
X
)
def
∼
H
X
=
H
1
(Δ
X
,
M
⊗
M
X
)
→
H
1
(Δ
X
,
M
[Gal(X
/X
)]
⊗
M
X
)
(l)
(l)
Then
the
“trace
map”
Tr
H
:
H
X
→
H
X
—
i.e.,
the
map
induced
by
the
morphism
of
coefficients
M
[Gal(X
/X
)]
M
that
maps
each
element
of
Gal(X
/X
)
to
1
—
factors
through
the
endomorphism
of
H
X
given
by
multiplication
by
l
n
[cf.
Remark
3.9.2
below].
[Indeed,
to
verify
(C),
we
recall
that
this
trace
map
Tr
H
is
well-known
to
be
dual
[via
Poincaré
duality
—
cf.,
e.g.,
[FK],
pp.
135-136]
to
the
pull-back
morphism;
∼
we
thus
conclude
that,
relative
to
the
natural
isomorphisms
H
X
→
Δ
ab
X
⊗
M
,
∼
ab
H
X
→
Δ
X
⊗
M
[arising
from
Poincaré
duality
—
cf.,
e.g.,
Proposition
1.3,
(ii)],
the
trace
map
corresponds
to
the
natural
morphism
ab
H
X
=
Δ
ab
X
⊗
M
→
Δ
X
⊗
M
=
H
X
induced
by
the
inclusion
Δ
X
⊆
Δ
X
—
hence,
by
the
definition
of
H
,
factors
through
the
endomorphism
of
H
X
given
by
multiplication
by
l
n
.
This
completes
the
proof
of
(C).]
Next,
let
us
suppose
that
we
have
been
given
morphisms
of
(S,
∅,
Σ)-admissible
coverings
of
X
∗
∗
→
X
→
X
→
X
X
→
X
∗
∗
∗
∗
and
write
U
S
⊆
X
,
U
S
∗
⊆
X
,
U
S
∗
⊆
X
for
the
open
subscheme
deter-
∗
∗
mined,
respectively,
by
the
complements
of
the
sets
S
,
S
,
S
of
closed
points
70
SHINICHI
MOCHIZUKI
∗
∗
of
X
,
X
,
X
that
lie
over
points
of
S.
Also,
let
us
assume
that
the
open
sub-
groups
Δ
X
⊆
Δ
X
,
Δ
X
∗
⊆
Δ
X
,
Δ
X
⊆
Δ
X
,
Δ
X
∗
⊆
Δ
X
arise,
respectively,
from
open
subgroups
H
=
l
n
·
Δ
ab
⊆
H
X
∗
∗
∗
H
=
l
n
·
Δ
ab
X
⊆
H
∗
∗
def
∗
=
l
n
∗
=
l
n
·
Δ
ab
⊆
Δ
ab
X
∗
X
∗
ab
·
Δ
ab
X
⊆
Δ
X
def
—
where
n
=
n
−
c,
n
=
n
−
c;
we
suppose
that
n
>
2c,
n
>
c
are
“sufficiently
large”
positive
integers,
to
be
chosen
below.
Then
we
wish
to
apply
the
theory
developed
above
[in
particular,
the
observations
(A),
(B),
(C)]
by
taking
“X
→
X
”
in
this
theory
to
be
various
subcoverings
of
X
→
X
.
Now
let
us
compute
the
cohomology
of
Π
X
via
the
Leray
spectral
sequence
associated
to
the
surjection
Π
X
Π
X
/Δ
X
∗
.
Suppose
that
c
has
been
chosen
so
∗
that
b
+
1
=
2c.
Then
by
applying
(A)
to
the
covering
“X
→
X
”
(respectively,
∗
“X
→
X
”),
we
conclude
that
Δ
X
∗
(respectively,
Δ
X
∗
)
acts
trivially
on
tor
New
tor
S
/S
∗
(b
+
1/b
+
2)
(respectively,
New
S
/S
∗
(b
+
1/b
+
2)).
Also,
it
follows
immediately
from
the
definitions
that
we
have
a
natural
Π
X
-equivariant
surjection
tor
New
tor
S
/S
∗
(b
+
1/b
+
2)
New
S
/S
∗
(b
+
1/b
+
2).
Now,
by
applying
(A)
to
the
∗
covering
“X
→
X
”
and
(C)
to
the
covering
“X
the
Π
X
-equivariant
natural
morphism
∗
→
X
∗
”,
we
conclude
that
tor
1
H
1
(Δ
X
∗
,
New
tor
∗
(b
+
1/b
+
2))
→
H
(Δ
∗
,
New
∗
(b
+
1/b
+
2))
X
S
/S
S
/S
[which
maps
the
image
of
η
S
to
the
image
of
η
S
!]
factors
through
a
“trace
∗
∗
→
X
”,
hence
in
particular,
through
the
map”
as
in
(C)
for
the
covering
“X
endomorphism
of
H
1
(Δ
X
∗
,
New
tor
∗
(b
+
1/b
+
2))
[a
module
whose
submodule
S
/S
∗
of
Π
X
-invariants
is
finite,
by
Proposition
3.4,
(i)]
given
by
multiplication
by
n
,
in
a
Π
X
-equivariant
fashion.
Thus,
by
taking
n
to
be
“sufficiently
large”,
we
conclude
that
the
image
of
η
S
in
H
1
(Δ
X
∗
,
New
tor
(b
+
1/b
+
2))
is
zero.
S
/S
∗
Now
I
claim
that
the
image
of
η
S
in
H
1
(Δ
X
,
New
tor
S
/S
(b
+
1/b
+
2))
[obtained
by
applying
the
surjection
tor
New
tor
S
(b
+
1/b
+
2))
New
S
/S
(b
+
1/b
+
2)
of
(A)
applied
to
the
covering
“X
→
X
”]
is
zero.
Indeed,
note
that
it
follows
im-
mediately
from
the
definitions
that
we
have
a
natural
surjection
New
tor
∗
(b+1/b+
S
/S
∗
2)
New
tor
S
/S
(b
+
1/b
+
2)
[induced,
in
effect,
by
the
inclusion
Gal(X
/X
)
→
Gal(X
/X
)].
Thus,
since
we
have
already
shown
that
the
image
of
η
S
in
the
co-
homology
module
H
1
(Δ
X
∗
,
New
tor
∗
(b+1/b+2))
is
zero,
it
follows
immediately
S
/S
1
that
the
image
of
η
S
in
H
(Δ
X
∗
,
New
tor
S
/S
(b
+
1/b
+
2))
is
zero,
hence
that
the
image
in
question
in
the
claim
arises
from
a
class
∈
H
1
(Gal(X
∗
Δ
X
∗
/X
),
(New
tor
)
S
/S
(b
+
1/b
+
2))
=
H
1
(Gal(X
∗
/X
),
New
tor
S
/S
(b
+
1/b
+
2))
=
0
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
71
[where
the
last
cohomology
module
vanishes
since,
by
(A)
applied
to
the
covering
∗
“X
→
X
”,
New
tor
S
/S
(b
+
1/b
+
2)
is
an
induced
Gal(X
/X
)-module].
This
completes
the
proof
of
the
claim.
Thus,
in
summary,
we
conclude
that
the
image
of
η
S
in
the
cohomology
module
H
1
(Π
X
,
New
tor
S
/S
(b
+
1/b
+
2))
arises
from
a
unique
class
in
∼
Δ
X
H
1
(Π
X
/Δ
X
,
(New
tor
)
→
H
1
(Π
X
/Δ
X
,
New
tor
S
/S
(b
+
1/b
+
2))
S
/S
(b
+
1/b
+
2))
which
maps
to
the
unique
class
in
H
1
(Π
X
/Δ
X
,
New
tor
S
(b
+
1/b
+
2))
[a
module
which
is
finite,
by
Proposition
3.4,
(i)]
that
gives
rise
to
η
S
via
a
mor-
phism
that
factors
through
the
endomorphism
given
by
multiplication
by
the
order
of
c
·
Gal(X
/X
)
[cf.
(A),
(B)
applied
to
the
covering
“X
→
X
”].
In
particular,
by
taking
n
to
be
“sufficiently
large”,
we
may
conclude
that
η
S
=
0,
as
desired.
That
is
to
say:
TOR≤b+1
This
completes
the
proof
that
the
two
inclusions
Π
≤b
differ
U
S
→
Π
U
S
TOR≤b+1
LIE≤b+
by
conjugation
by
an
element
of
Ker(Π
U
S
Π
U
S
).
In
particular,
the
proof
of
assertions
(i),
(ii)
is
complete.
Next,
we
consider
assertion
(iii).
First,
let
us
observe
that
when
b
=
1,
assertion
(iii)
follows
immediately
from
[the
“pro-l
version”
of
the
argument
applied
to
prove]
Proposition
2.6,
(i)
[cf.
the
discussion
preceding
Proposition
3.7].
Thus,
in
the
remainder
of
the
proof
of
assertion
(iii),
we
assume
that
b
≥
2.
Note
that
since
the
≤b+1
Π
≤b
,
elements
of
Ker(Π
≤b+1
U
S
U
S
)
manifestly
commute
with
the
elements
of
I
x
∗
it
follows
from
conditions
(a),
(b),
the
fact
that
b
≥
2,
and
the
assumption
that
[csp]
[cf.
the
proof
S
=
S
∗
is
of
cardinality
one
that
β
induces
the
identity
on
Π
≤b+1
U
S
of
assertion
(ii)
above].
Thus,
to
complete
the
proof
of
assertion
(iii),
it
suffices
to
show
that
the
compatible
system
of
classes
λ
S
∈
H
1
(Π
X
,
New
S
(b
+
1/b
+
2))
determined
by
β
[cf.
Proposition
3.7,
(i);
3.3,
(iv)]
vanishes.
Note
that
since
(l)
(Δ
X
)
ab
is
of
“weight
≤
1”,
and
New
S
(b
+
1/b
+
2)
is
of
“weight
b
+
1
≥
3”
[cf.
Proposition
3.4,
(i)],
it
follows
immediately
from
the
Leray
spectral
sequence
for
Π
X
G
k
that
we
have
a
natural
isomorphism
∼
H
1
(G
k
,
(New
S
(b
+
1/b
+
2))
Δ
X
)
→
H
1
(Π
X
,
New
S
(b
+
1/b
+
2))
[where
the
superscript
“Δ
X
”
denotes
the
Δ
X
-invariants]
and
that
the
module
H
1
(G
k
,
(New
S
(b
+
1/b
+
2))
Δ
X
)
is
finite.
Thus,
to
show
that
the
λ
S
=
0,
it
suffices
to
show
that
the
inverse
limit
Δ
X
lim
←−
(New
S
(b
+
1/b
+
2))
X
72
SHINICHI
MOCHIZUKI
[where
X
,
S
are
as
described
in
the
proof
of
assertions
(i),
(ii)]
is
zero.
But
this
follows
from
observation
(B)
of
the
proof
of
assertions
(i),
(ii).
This
completes
the
proof
of
assertion
(iii).
,
it
suf-
Next,
we
consider
assertion
(iv).
In
light
of
the
definition
of
Π
TOR≤b+1
U
S
fices
to
show
that
any
compatible
system
of
D
X
-invariant
[relative
to
the
diagonal
action
of
D
X
]
classes
κ
S
∈
New
tor
S
(b
+
1/b
+
2)
[where
X
,
S
are
as
described
in
the
proof
of
assertions
(i),
(ii)]
lies
in
the
image
of
I
x
∗
[U
S
]
⊗
(Q
l
/Z
l
)
if
b
=
1
and
vanishes
if
b
≥
2.
[Here,
we
note
that
since
New
tor
S
(b
+
1/b
+
2)
=
0
when
b
is
even,
we
may
assume
without
loss
of
generality
that
b
is
odd.]
To
do
this,
let
X
,
X
,
S
,
S
be
as
in
(A),
(B).
Now
we
would
like
to
apply
the
theory
of
the
Appendix
[cf.,
especially,
Theorem
A.5]
to
the
present
situation.
To
do
this,
it
is
necessary
to
specify
the
data
“(i),
(ii),
(iii),
(vi),
(vii),
(viii),
(ix),
(x),
(xi),
(xii)”
[cf.
the
discussion
of
the
Appendix]
to
which
this
theory
is
to
be
applied.
We
take
the
“d”
of
Theorem
A.5
to
be
such
that
2d
=
b
+
1
[so
the
fact
that
b
is
odd
implies
that
d
≥
2
whenever
b
≥
2]
and
the
prime
number
“l”
of
“(i)”
to
be
the
prime
number
l
of
the
present
discussion.
We
take
the
profinite
group
“Δ”
of
“(ii)”
to
be
the
quotient
of
the
group
Δ
X
by
the
kernel
of
the
quotient
ab
(Δ
X
⊇)
Δ
X
Δ
ab
X
Δ
X
⊗
Z
l
;
this
group
“Δ”
surjects
onto
Δ
X
/Δ
X
,
which
we
take
to
be
the
quotient
group
“G”
of
“(ii)”,
with
kernel
Δ
ab
X
⊗
Z
l
,
which
we
take
to
be
the
subgroup
“V
”
of
“(ii)”.
Here,
we
recall
that
the
condition
of
“(ii),
(c)”
concerning
the
regular
representation
follows
immediately
from
[Milne],
p.
187,
Corollary
2.8
[cf.
also
[Milne],
p.
187,
Remark
2.9],
in
light
of
our
assumption
that
X
is
proper
hyperbolic,
hence
of
genus
≥
2.
We
take
the
profinite
group
“Γ”
of
“(ix)”
to
be
the
image
G
k
⊆
G
k
of
Π
X
in
G
k
[so
“Γ”
acts
naturally
on
“Δ”,
“G”,
“H”].
Thus,
“Δ
Γ
”
may
be
thought
of
as
a
quotient
of
Π
X
×
G
k
G
k
,
hence
also
as
a
quotient
of
D
X
×
G
k
G
k
.
Note
that
by
consideration
of
“weights”,
it
follows
that
G
k
(New
tor
S
(b
+
1/b
+
2))
is
finite,
hence
annihilated
by
some
finite
power
of
l,
which
we
take
to
be
the
number
“N
”
of
“(iii)”.
We
take
the
covering
X
→
X
of
(A),
(B)
to
be
any
(S,
∅,
Σ)-
admissible
covering
such
that
the
resulting
covering
X
k
→
X
k
is
the
covering
determined
by
the
resulting
subgroup
“l
n
·
V
⊆
Δ”
[cf.
the
statement
of
Theorem
A.5],
so
“J”
may
be
identified
with
Gal(X
/X
).
Next,
we
take
the
“G-torsor
E
G
”
of
“(vi)”
to
be
S
and
the
“H-torsor
E
H
”
of
“(vii)”
to
be
S
;
thus,
the
natural
surjection
S
S
determines
a
surjection
“E
H
E
G
”
as
in
“(viii)”.
Note
that
S
(respectively,
S
)
may
be
thought
of
as
a
Δ
U
S
-orbit
(respectively,
Δ
U
S
-
orbit)
[via
the
action
by
conjugation]
of
the
conjugacy
class
of
subgroups
of
Δ
U
S
determined
by
I
x
∗
[U
S
]
⊆
Δ
U
S
.
In
particular,
it
follows
that
the
particular
member
of
this
conjugacy
class
constituted
by
the
subgroup
I
x
∗
[U
S
]
⊆
Δ
U
S
determines
a
particular
element
e
H
∈
E
H
(respectively,
e
G
∈
E
G
)
as
in
“(xi)”.
Moreover,
the
diagonal
action
of
D
X
—
hence
also
of
D
X
×
G
k
G
k
⊆
D
X
—
on
Δ
U
S
determines
an
action
of
D
X
×
G
k
G
k
⊆
D
X
on
E
H
,
E
G
that
fixes
e
H
,
e
G
,
and
[as
is
easily
verified]
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
73
factors
through
the
quotient
“Δ
Γ
”
of
D
X
×
G
k
G
k
Π
X
×
G
k
G
k
;
in
particular,
we
obtain
continuous
actions
of
“Δ
Γ
”
on
“E
G
”,
“E
H
”
as
in
“(x)”.
Finally,
we
take
(l)
the
“Γ-module
Λ”
of
“(xii)”
to
be
the
d-th
tensor
power
of
M
X
⊗
(Q
l
/Z
l
).
This
completes
the
specification
of
the
data
necessary
to
apply
Theorem
A.5.
Thus,
by
applying
Theorem
A.5
to
the
composite
of
the
second
and
third
surjections
in
the
factorization
of
(A),
we
conclude
that
since
κ
S
is
D
X
-invariant,
it
follows
that
κ
S
∈
New
tor
S
(b
+
1/b
+
2)
G
k
=
0
when
b
≥
2
and
to
maps
to
an
element
[i.e.,
κ
S
]
of
N
·
New
tor
S
(b
+
1/b
+
2)
an
element
[i.e.,
κ
S
]
in
the
image
of
I
x
∗
[U
S
]
⊗
(Q
l
/Z
l
)
when
b
=
1.
This
completes
the
proof
of
assertion
(iv).
Finally,
we
consider
assertion
(v).
It
is
immediate
from
the
definitions
that
the
various
quotients
in
question
are
cuspidally
pro-l.
That
these
quotients
are
the
maximal
cuspidally
pro-l
quotients
follows
from
the
construction
of
Δ
≤∞
U
S
and
the
(l)
(l)
easily
verified
fact
that
each
Δ
U
injects
into
Lin(Δ
U
(1/∞))(Q
l
).
Finally,
the
S
S
asserted
slimness
follows
from
the
fact
that
the
profinite
groups
in
question
may
be
written
as
inverse
limits
of
profinite
groups
that
admit
normal
open
subgroups
(l)
(l)
(l)
(l)
[with
trivial
centralizers]
—
namely,
“Δ
U
”,
“(Π
U
)
†
”,
“Δ
U
X
×X
”,
“(Π
U
X
×X
)
†
”
S
S
—
which
are
slim,
by
Proposition
1.8,
(i),
(iii)
[which
implies
that
the
quotients
(l)
(l)
(l)
(l)
Δ
U
X
×X
Δ
X
,
(Π
U
X
×X
)
†
(Π
X
)
†
,
as
well
as
the
kernels
of
these
quotients,
are
slim].
Remark
3.9.1.
Proposition
3.9,
(iii),
may
be
regarded
as
a
“higher
order,
pro-l
analogue”
of
Proposition
2.6,
(i).
Remark
3.9.2.
It
is
important
to
note
that
if
one
omits
[as
was,
mistakenly,
done
in
an
earlier
version
of
this
paper]
the
hypothesis
that
S
0
=
∅,
then
it
no
longer
holds
that
the
image
of
the
trace
map
“Tr
H
:
H
X
→
H
X
”
[appearing
in
the
proof
of
Proposition
3.9,
(i),
(ii)]
lies
in
l
n
·
H
X
.
Indeed,
this
phenomenon
may
be
understood
by
considering
the
trace
map
on
first
étale
cohomology
modules
with
Z
l
-coefficients
associated
to
the
l
n
-th
power
map
G
m
→
G
m
on
the
multiplicative
group
G
m
over
k
—
a
map
which,
as
an
easy
computation
reveals,
is
surjective.
We
are
now
ready
to
prove
the
main
technical
result
of
the
present
§3:
Theorem
3.10.
(Reconstruction
of
Maximal
Cuspidally
Pro-l
Exten-
sions)
Let
X,
Y
be
proper
hyperbolic
curves
over
a
finite
field;
denote
the
base
fields
of
X,
Y
by
k
X
,
k
Y
,
respectively.
Suppose
further
that
we
have
been
def
def
def
given
points
x
∗
∈
X(k
X
),
y
∗
∈
Y
(k
Y
);
write
S
=
{x
∗
},
T
=
{y
∗
}
U
S
=
X\S,
def
V
T
=
Y
\T
.
Let
Σ
be
a
set
of
prime
numbers
that
contains
at
least
one
prime
number
that
is
invertible
in
k
X
,
k
Y
;
thus,
Σ
determines
various
quotients
Π
U
S
,
74
SHINICHI
MOCHIZUKI
Π
X
,
Π
U
X×X
,
Π
X×X
,
Π
V
T
,
Π
Y
,
Π
U
Y
×Y
,
Π
Y
×Y
[cf.
Proposition
1.8,
(iii);
the
dis-
cussion
preceding
Proposition
1.6]
of
the
étale
fundamental
groups
of
U
S
,
X,
U
X×X
,
X
×
X,
V
T
,
Y
,
U
Y
×Y
,
Y
×
Y
,
respectively.
Also,
we
write
Π
X
G
k
X
,
Π
Y
G
k
Y
for
the
quotients
determined
by
the
respective
absolute
Galois
groups
of
k
X
,
k
Y
.
Let
∼
α
:
Π
X
→
Π
Y
be
a
Frobenius-preserving
[hence
also
quasi-point-theoretic
—
cf.
Remark
1.18.2]
isomorphism
of
profinite
groups
that
maps
the
decomposition
group
of
x
∗
in
Π
X
[which
is
well-defined
up
to
conjugation]
to
the
decomposition
group
of
y
∗
in
Π
Y
[which
is
well-defined
up
to
conjugation].
Then
for
each
prime
l
∈
Σ
such
that
l
=
p,
there
exist
commutative
diagrams
Π
≤∞
U
S
⏐
⏐
∞
−→
Π
X
−→
α
α
Π
≤∞
V
T
⏐
⏐
Π
≤∞
U
X×X
⏐
⏐
Π
Y
Π
X×X
α
×
∞
−→
α×α
−→
Π
≤∞
U
Y
×Y
⏐
⏐
Π
Y
×Y
≤∞
≤∞
≤∞
—
in
which
Π
U
S
Π
≤∞
U
S
,
Π
U
X×X
Π
U
X×X
,
Π
V
T
Π
V
T
,
Π
U
Y
×Y
Π
U
Y
×Y
are
the
maximal
cuspidally
pro-l
quotients
[cf.
Proposition
3.9,
(v)];
Π
X×X
∼
=
Π
×
Π
;
the
vertical
arrows
are
the
natural
surjections;
Π
X
×
G
kX
Π
X
,
Π
Y
×Y
∼
=
Y
G
kY
Y
are
isomorphisms,
well-defined
up
to
composition
with
a
cuspidally
inner
α
∞
,
α
×
∞
automorphism,
that
are
compatible,
relative
to
the
natural
surjections
c-ab,l
Π
≤∞
U
S
Π
U
S
;
c-ab,l
Π
≤∞
U
X×X
Π
U
X×X
;
c-ab,l
Π
≤∞
;
V
T
Π
V
T
c-ab,l
Π
≤∞
U
Y
×Y
Π
U
Y
×Y
—
where
we
use
the
superscript
“c-ab,
l”
to
denote
the
respective
maximal
cusp-
idally
pro-l
abelian
quotients
—
with
the
isomorphisms
∼
c-ab
Π
c-ab
U
S
→
Π
V
T
;
∼
c-ab
Π
c-ab
U
X×X
→
Π
U
Y
×Y
of
Theorem
2.5,
(i);
Theorem
1.16,
(iii),
respectively.
Moreover,
α
∞
(respectively,
α
×
∞
)
is
compatible,
up
cuspidally
inner
automorphisms,
with
the
decomposition
≤∞
groups
of
x
∗
,
y
∗
in
Π
≤∞
U
S
,
Π
V
T
(respectively,
with
the
images
of
the
decomposition
≤∞
groups
D
X
,
D
Y
in
Π
≤∞
U
X×X
,
Π
U
Y
×Y
).
Finally,
this
condition
of
“compatibility
with
decomposition
groups”,
together
with
the
condition
of
making
the
above
diagrams
commute,
uniquely
determine
the
isomorphisms
α
∞
,
α
×
∞
,
up
to
composition
with
×
a
cuspidally
inner
automorphism;
in
particular,
α
∞
is
compatible,
up
to
compo-
sition
with
a
cuspidally
inner
automorphism,
with
the
automorphisms
of
Π
≤∞
U
X×X
,
Π
≤∞
U
Y
×Y
given
by
switching
the
two
factors.
Proof.
First,
let
us
consider
the
isomorphism
[i.e.,
more
precisely:
a
specific
mem-
ber
of
the
cuspidally
inner
equivalence
class
of
isomorphisms]
∼
c-ab,l
α
c-ab,l
:
Π
c-ab,l
U
X×X
→
Π
U
Y
×Y
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
75
∼
c-ab
arising
from
the
isomorphism
Π
c-ab
U
X×X
→
Π
U
Y
×Y
of
Theorem
1.16,
(iii).
Recall
that
since
α
is
Frobenius-preserving,
it
is
quasi-point-theoretic
[cf.
Remark
1.18.2],
and
that
α
c-ab,l
is
compatible
with
the
images
of
D
X
,
D
Y
,
which
we
denote
by
(l)
(l)
D
X
,
D
Y
.
Thus,
we
may
assume
without
loss
of
generality
that
our
choices
of
decomposition
groups
D
x
∗
[U
S
]
⊆
Π
U
S
,
D
y
∗
[V
T
]
⊆
Π
V
T
,
as
well
as
our
choices
of
c-ab,l
splittings
G
k
X
→
D
x
∗
[U
S
],
G
k
Y
→
D
y
∗
[V
T
],
have
images
in
Π
c-ab,l
U
X×X
,
Π
U
Y
×Y
that
⊆
Π
c-ab,l
correspond
via
α
c-ab,l
.
In
particular,
it
follows
that
α
c-ab,l
maps
Π
c-ab,l
U
S
U
X×X
isomorphically
onto
Π
c-ab,l
⊆
Π
c-ab,l
V
T
U
Y
×Y
.
In
the
following
argument,
let
us
identify
the
“Lin
U
S
(1/∞)”,
“Lin
X
(1/∞)”
portions
of
Δ
Lie
U
S
with
the
[completions,
relative
to
the
natural
filtration
topology,
of
the]
corresponding
graded
objects
“Gr
Q
l
(−)(1/∞)”
via
the
Galois
invariant
split-
tings
of
Proposition
3.4,
(ii),
and
similarly
for
V
T
.
Then,
in
light
of
our
assumption
that
α
is
Frobenius-preserving,
it
follows
immediately
from
the
naturality
of
our
con-
structions
[cf.,
especially,
Proposition
3.4,
(iii)]
that
α
induces,
for
each
Z
b
≥
1,
compatible
isomorphisms
∼
LIE
α
LIE
:
Π
LIE
U
S
→
Π
V
T
;
∼
α
LIE≤b
:
Π
LIE≤b
→
Π
LIE≤b
U
S
V
T
which
are,
moreover,
compatible
[with
respect
to
the
natural
projections
to
Π
X
,
Π
Y
]
”
with
the
isomorphism
α.
Moreover,
it
follows
from
the
construction
of
“Π
LIE≤b
(−)
that
the
latter
displayed
isomorphism
maps
D
x
LIE≤b
⊆
Π
LIE≤b
bijectively
onto
U
S
∗
LIE≤b
LIE≤b
LIE≤b
∼
D
y
∗
⊆
Π
V
T
,
and
that
the
resulting
isomorphism
D
x
∗
→
D
y
LIE≤b
induces
∗
an
isomorphism
∼
D
x
≤b
→
D
y
≤b
∗
∗
which
is
compatible
[again
by
construction!]
with
the
respective
Frobenius
elements
“F
k
”
on
either
side.
Next,
let
us
observe
that
since
the
isomorphism
α
c-ab,l
induces
an
isomorphism
∼
Π
c-ab,l
→
Π
c-ab,l
that
is
compatible
with
the
images
of
the
decompositions
groups
U
S
V
T
D
x
∗
[U
S
],
D
y
∗
[V
T
]
and
Frobenius
elements
in
these
decomposition
groups,
it
fol-
lows
immediately
that
for
corresponding
[i.e.,
via
α]
(S,
∅,
Σ)-,
(T,
∅,
Σ)-admissible
coverings
X
→
X,
Y
→
Y
[which
induce
coverings
U
S
→
U
S
,
V
T
→
V
T
],
∼
α
c-ab,l
induces
an
isomorphism
Δ
Lie≤2
→
Δ
Lie≤2
which
is
compatible
with
α
LIE≤2
.
V
T
U
S
,
Δ
Lie≤2
are
not
center-free,
the
semi-direct
products
Moreover,
although
Δ
Lie≤2
V
T
U
S
Δ
Lie≤2
H
X
,
Δ
Lie≤2
H
Y
are
easily
seen
to
be
center-free
[cf.
Proposition
1.8,
V
T
U
S
(i)],
for
arbitrary
open
subgroups
H
X
⊆
G
†
k
X
,
H
Y
⊆
G
†
k
Y
[where
the
daggers
are
as
in
Proposition
1.8,
(i)]
that
correspond
via
α.
Since
Π
LIE≤2
(respectively,
Π
LIE≤2
)
U
S
V
T
is
an
inverse
limit
of
topological
groups
that
admit
normal
closed
subgroups
of
the
H
X
(respectively,
Δ
Lie≤2
H
Y
),
we
thus
conclude
[by
applying
the
form
Δ
Lie≤2
V
T
U
S
extension
“1
→
(−)
→
Aut(−)
→
Out(−)
→
1”
of
§0
to
these
normal
closed
sub-
∼
→
Π
c-ab,l
induced
by
α
c-ab,l
is
compatible
—
groups]
that
the
isomorphism
Π
c-ab,l
U
S
V
T
relative
to
the
natural
inclusions
∼
LIE≤2
Π
c-ab,l
→
Π
≤2
;
U
S
→
Π
U
S
U
S
∼
LIE≤2
Π
c-ab,l
→
Π
≤2
V
T
→
Π
V
T
V
T
76
SHINICHI
MOCHIZUKI
∼
[cf.
the
discussion
preceding
Proposition
3.7]
—
with
α
LIE≤2
:
Π
LIE≤2
→
Π
LIE≤2
.
U
S
V
T
In
fact,
since
3
is
odd,
it
follows
immediately
from
the
definitions
that
the
modules
“New
Q
S
(3/4)”
vanish,
hence
[cf.
Definition
3.5,
(ii)]
that
we
have
an
iso-
LIE≤3
∼
→
Π
LIE≤2+
,
which
implies
[cf.
Proposition
3.7,
(i)]
that
we
have
morphism
Π
U
S
U
S
≤3
∼
≤2
an
isomorphism
Π
U
S
→
Π
U
S
[and
similarly
for
V
T
].
Thus,
by
Proposition
3.7,
(iii),
∼
→
Π
c-ab,l
induced
by
α
c-ab,l
is
compatible
—
it
follows
that
the
isomorphism
Π
c-ab,l
U
S
V
T
relative
to
the
natural
inclusions
∼
LIE≤3
→
Π
≤3
;
Π
c-ab,l
U
S
U
S
→
Π
U
S
∼
LIE≤3
Π
c-ab,l
→
Π
≤3
V
T
V
T
→
Π
V
T
∼
—
with
α
LIE≤3
:
Π
LIE≤3
→
Π
LIE≤3
.
U
S
V
T
LIE
Next,
let
us
observe
that
the
diagonal
actions
of
D
X
,
D
Y
on
Π
LIE
U
S
,
Π
V
T
clearly
(l)
(l)
(l)
(l)
(l)
(l)
factor
through
D
X
,
D
Y
[hence
determine
“diagonal
actions”
of
D
X
,
D
Y
on
Π
LIE
U
S
,
].
Moreover,
by
what
we
have
already
shown
concerning
the
compatibility
of
Π
LIE
V
T
α
LIE≤3
with
α
c-ab,l
[cf.
also
the
compatibility
of
α
c-ab,l
with
D
X
,
D
Y
]
and
the
compatibility
of
α
c-ab,l
with
the
decomposition
groups
D
x
∗
[U
S
],
D
y
∗
[V
T
],
it
follows
[cf.
Remarks
3.8.1,
3.8.2]
that
the
conditions
(a),
(b),
(c)
of
Proposition
3.8
are
compatible
with
α
LIE
,
hence
that
α
LIE
is
compatible
with
the
diagonal
actions
of
(l)
(l)
(l)
∼
(l)
LIE
D
X
,
D
Y
on
Π
LIE
U
S
,
Π
V
T
[relative
to
the
isomorphism
D
X
→
D
Y
induced
by
α
c-ab,l
].
≤b
Now
I
claim
that
the
isomorphism
α
LIE≤b
maps
Π
≤b
U
S
bijectively
onto
Π
V
T
,
thus
inducing
a
compatible
inverse
system
[parametrized
by
b]
of
isomorphisms
∼
≤b
α
≤b
:
Π
≤b
U
S
→
Π
V
T
≤b
that
are
compatible
[with
respect
to
the
natural
projections
Π
≤b
U
S
Π
X
,
Π
V
T
Π
Y
]
with
α.
To
verify
this
claim,
we
apply
induction
on
b.
The
case
b
=
1
is
vacuous;
the
case
b
=
2
follows
from
what
we
have
already
shown
concerning
the
compatibility
of
α
LIE≤2
with
α
c-ab,l
.
Thus,
we
assume
that
b
≥
2,
and
that
the
claim
has
been
verified
for
“b”
that
are
≤
the
b
under
consideration.
Now
observe
that
by
Propositions
3.7,
(iii);
3.9,
(ii),
it
follows
that
the
isomor-
phism
LIE≤b+1
∼
Π
U
→
Π
V
LIE≤b+1
S
T
maps
Π
≤b+1
bijectively
onto
a
Ker(Π
LIE≤b+1
Π
V
LIE≤b+
)-conjugate
of
Π
≤b+1
.
U
S
V
T
V
T
T
LIE≤b+1
In
particular,
by
conjugating
by
an
appropriate
element
γ
∈
Ker(Π
V
T
LIE≤b+
Π
V
T
),
we
obtain
an
isomorphism
∼
β
b+1
:
Π
≤b+1
→
Π
≤b+1
U
S
V
T
that
is
compatible
with
α
≤b
and,
moreover,
[since
γ
commutes
with
I
y
≤b+1
]
maps
∗
≤b+1
≤b+1
bijectively
onto
I
y
∗
.
Note
that
by
Propositions
3.7,
(i);
3.9,
(iii),
it
follows
I
x
∗
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
77
that
the
choice
of
γ
is
unique,
modulo
Ker(Π
≤b+1
Π
≤b+
V
T
V
T
).
In
particular,
the
TOR≤b+1
TOR≤b+1
image
δ
∈
Π
V
T
of
γ
in
Π
V
T
is
uniquely
determined.
(l)
On
the
other
hand,
since
α
LIE
is
compatible
with
the
diagonal
actions
of
D
X
,
(l)
LIE
D
Y
on
Π
LIE
U
S
,
Π
V
T
,
it
follows
immediately,
by
“transport
of
structure”,
that
δ
is
(l)
fixed
by
the
diagonal
action
of
D
Y
.
But,
by
Proposition
3.9,
(iv),
this
implies
that
δ
=
0.
This
completes
the
proof
of
the
claim.
∼
≤∞
Thus,
we
obtain
an
isomorphism
α
∞
:
Π
≤∞
U
S
→
Π
V
T
as
in
the
statement
of
≤∞
Theorem
3.10.
Next,
let
us
recall
that
Δ
≤∞
U
S
,
Δ
V
T
are
slim
[cf.
Proposition
3.9,
(v)].
Thus,
since
this
isomorphism
α
∞
is
compatible
with
the
diagonal
actions
of
∼
(l)
(l)
≤∞
D
X
,
D
Y
,
we
may
apply
the
isomorphism
Aut(Δ
≤∞
U
S
)
→
Aut(Δ
V
T
)
induced
by
α
∞
to
obtain
—
i.e.,
by
pulling
back
the
extension
≤∞
≤∞
1
→
Δ
≤∞
U
S
→
Aut(Δ
U
S
)
→
Out(Δ
U
S
)
→
1
[cf.
§0]
via
the
homomorphism
(D
X
)
Π
X
→
Out(Δ
≤∞
U
S
)
(l)
×
arising
from
the
diagonal
action
[and
similarly
for
Δ
≤∞
V
T
]
—
an
isomorphism
α
∞
:
∼
≤∞
Π
≤∞
U
X×X
→
Π
U
Y
×Y
as
in
the
statement
of
Theorem
3.10.
Here,
we
note
that
the
“cuspidally
inner
indeterminacy”
of
α
∞
,
α
×
∞
that
is
referred
to
in
the
statement
of
Theorem
3.10
arises
from
the
“cuspidally
inner
indeterminacy”
in
the
choice
of
corresponding
decomposition
groups
D
x
∗
[U
S
],
D
y
∗
[V
T
]
[more
precisely:
the
images
≤∞
c-ab,l
c-ab,l
].
Finally,
we
ob-
of
these
groups
in
Π
≤∞
U
S
,
Π
V
T
,
as
opposed
to
just
in
Π
U
S
,
Π
V
T
serve
that
the
asserted
uniqueness
follows
immediately
by
considering
eigenspaces
relative
to
the
Frobenius
actions
[cf.
Proposition
3.4,
(ii)],
together
with
the
con-
struction
of
the
isomorphism
α
LIE
[cf.
also
Propositions
1.15,
(i);
2.6,
(i)].
Remark
3.10.1.
The
argument
of
the
proof
of
Theorem
3.10
involving
Proposi-
tion
3.9,
(iv),
may
be
regarded
as
a
sort
of
“higher
order
analogue”
of
the
argument
applied
in
the
proof
of
Theorem
1.16,
(iii),
involving
Lemma
1.11;
Proposition
1.12,
(v).
Remark
3.10.2.
At
first
glance,
it
may
appear
that
the
portion
of
Theorem
3.10
concerning
α
×
∞
may
only
be
concluded
when
X(k
X
),
Y
(k
Y
)
are
nonempty.
In
fact,
≤∞
†
†
however,
since
(Π
≤∞
U
X×X
)
,
(Π
U
Y
×Y
)
are
slim
[cf.
Proposition
3.9,
(v)],
it
follows
that
the
portion
of
Theorem
3.10
concerning
α
×
∞
may
be
concluded
even
without
assuming
that
X(k
X
),
Y
(k
Y
)
are
nonempty,
by
applying
Theorem
3.10
after
passing
to
corresponding
[via
α]
finite
extensions
of
k
X
,
k
Y
[cf.
Remark
1.10.1].
Remark
3.10.3.
It
seems
reasonable
to
expect
that,
when,
say,
Σ
=
{l},
the
techniques
applied
in
the
proof
of
Theorem
3.10,
together
with
the
theory
of
[Mtm],
78
SHINICHI
MOCHIZUKI
should
allow
one
to
reconstruct
the
[geometrically
pro-Σ]
étale
fundamental
groups
of
the
various
configuration
spaces
[i.e.,
finite
products
of
copies
of
X
over
k
X
,
with
the
various
diagonals
removed]
“group-theoretically”
from
Π
X
[under,
say,
an
appropriate
hypothesis
of
“Frobenius-preservation”
as
in
Theorem
3.10].
This
topic,
however,
lies
beyond
the
scope
of
the
present
paper.
Remark
3.10.4.
If
the
“cuspidalization
of
configuration
spaces”
[cf.
Remark
3.10.3]
can
be
achieved,
then
it
seems
likely
that
by
applying
an
appropriate
“spe-
cialization”
operation,
it
should
be
possible
to
generalize
Theorem
3.10
to
the
case
where
S,
T
are
subsets
of
arbitrary
finite
cardinality.
Remark
3.10.5.
One
essential
portion
of
the
proof
of
Theorem
3.10
is
the
Galois
invariant
splitting
of
Proposition
3.4,
(ii).
Although
it
does
not
appear
likely
that
such
a
splitting
exists
in
the
case
of
a
nonarchimedean
local
base
field
[cf.,
e.g.,
the
theory
of
[Mzk4]],
it
would
be
interesting
to
investigate
the
extent
to
which
a
result
such
as
Theorem
3.10
may
be
generalized
to
the
nonarchimedean
local
case,
perhaps
by
making
use
of
some
sort
of
splitting
such
as
the
Hodge-Tate
decomposition,
or
a
splitting
that
arises
via
crystalline
methods.
In
the
context
of
absolute
anabelian
geometry
over
nonarchimedean
local
fields,
however,
such
p-adic
Hodge-theoretic
splittings
might
not
be
available,
since
the
isomorphism
class
of
the
Galois
module
“C
p
”
is
not
preserved
by
arbitrary
automorphisms
of
the
absolute
Galois
group
of
a
nonarchimedean
local
field
[cf.
the
theory
of
[Mzk3]].
The
development
of
the
theory
underlying
Theorem
3.10
was
motivated
by
the
following
important
consequence:
Corollary
3.11.
(Total
Global
Green-compatibility)
In
the
situation
of
Theorem
1.16,
(iii)
[in
the
finite
field
case],
suppose
further
that
Σ
†
=
Primes
†
,
and
that
X,
Y
are
Σ-separated
[which
implies
that
α
is
Frobenius-preserving
and
point-theoretic
—
cf.
Remarks
1.18.1,
1.18.2].
Then
the
isomorphism
α
is
totally
globally
Green-compatible.
Proof.
Indeed,
we
may
apply
Theorem
3.10
to
the
isomorphism
α
of
Theorem
1.16,
(iii),
and
arbitrary
choices
of
sets
of
cardinality
one
S
=
{x
∗
},
T
=
{y
∗
}
that
correspond
via
α.
Let
l
∈
Σ
†
.
Then
let
us
observe
that
the
quotient
Π
U
S
Π
≤∞
U
S
satisfies
the
following
property:
If
Π
U
S
Q
is
a
finite
quotient
of
Π
U
S
such
that
for
some
quotient
Q
Q
whose
kernel
has
order
a
power
of
l,
Π
U
S
Q
factors
through
Π
U
S
≤∞
Π
≤∞
U
S
,
then
Π
U
S
Q
also
factors
through
Π
U
S
Π
U
S
.
A
similar
statement
holds
for
the
quotient
Π
V
T
Π
≤∞
V
T
.
In
light
of
this
observation,
†
†
together
with
our
assumption
that
Σ
=
Primes
[which
implies
that
α
is
Frobenius-
preserving],
it
follows
that
the
reasoning
of
[Tama],
Corollary
2.10,
Proposition
3.8
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
79
[cf.
also
Remark
1.18.2
of
the
present
paper],
may
be
applied
to
the
isomorphism
∼
≤∞
α
∞
:
Π
≤∞
U
S
→
Π
V
T
of
Theorem
3.10
to
conclude
that
the
isomorphism
α
∞
maps
the
set
of
decompo-
sition
subgroups
of
the
domain
bijectively
onto
the
set
of
decomposition
subgroups
of
the
codomain.
On
the
other
hand,
sorting
through
the
definitions,
the
datum
of
the
lifting
of
a
decomposition
group
of
Π
X
,
Π
Y
corresponding
to
a
point
that
does
not
belong
to
S,
T
to
a
[noncuspidal]
decomposition
group
of
the
domain
or
codomain
of
c-ab,l
α
∞
determines,
by
projection
to
Π
c-ab,l
,
the
l-adic
portion
of
the
Green’s
U
S
,
Π
V
T
trivialization
associated
to
this
point
and
the
unique
point
of
S
or
T
.
Since
l
is
an
arbitrary
element
of
Σ
†
=
Primes
†
,
and
the
points
x
∗
,
y
∗
are
arbitrary
points
that
correspond
via
α,
this
shows
that
α
is
globally
Green-compatible.
That
α
is
totally
globally
Green-compatible
follows
by
applying
this
argument
to
the
isomorphism
induced
by
α
between
open
subgroups
of
Π
X
,
Π
Y
.
Theorem
3.12.
(The
Grothendieck
Conjecture
for
Proper
Hyperbolic
Curves
over
Finite
Fields)
Let
X,
Y
be
proper
hyperbolic
curves
over
a
finite
field;
denote
the
base
fields
of
X,
Y
by
k
X
,
k
Y
,
respectively.
Write
Π
X
,
Π
Y
for
the
étale
fundamental
groups
of
X,
Y
,
respectively.
Let
∼
α
:
Π
X
→
Π
Y
be
an
isomorphism
of
profinite
groups.
Then
α
arises
from
a
uniquely
deter-
mined
commutative
diagram
of
schemes
∼
X
⏐
⏐
→
X
→
∼
Y
⏐
⏐
Y
in
which
the
horizontal
arrows
are
isomorphisms;
the
vertical
arrows
are
the
pro-
finite
étale
universal
coverings
determined
by
the
profinite
groups
Π
X
,
Π
Y
.
Proof.
Theorem
3.12
follows
formally
from
Corollaries
2.7,
3.11;
Remarks
1.18.1,
1.18.2;
Proposition
2.3,
(ii).
80
SHINICHI
MOCHIZUKI
Appendix:
Free
Lie
Algebras
In
this
present
Appendix,
we
discuss
various
elementary
facts
concerning
free
Lie
algebras
that
are
necessary
in
§3.
In
particular,
we
develop
a
sort
of
“higher
order
analogue”
of
the
theory
developed
in
Lemma
1.11.
Proposition
A.1.
(Free
Lie
Algebras)
Let
R
be
a
commutative
ring
with
unity;
V
a
finitely
generated
free
R-module.
Write
Lie
R
(V
)
for
the
free
Lie
algebra
over
R
associated
to
V
;
for
Z
b
≥
1,
denote
by
Lie
bR
(V
)
⊆
Lie
R
(V
)
the
R-submodule
generated
by
the
“alternants
of
degree
b”
[cf.
[Bour],
Chapter
II,
§2.6].
Also,
we
shall
denote
by
U
R
(V
)
the
enveloping
algebra
of
Lie
R
(V
).
[Thus,
we
have
a
natural
inclusion
Lie
R
(V
)
→
U
R
(V
).]
Then:
(i)
Each
Lie
bR
(V
)
is
a
finitely
generated
free
R-module.
Moreover,
there
is
∼
a
natural
isomorphism
V
→
Lie
1
R
(V
).
(ii)
Let
v
∈
V
be
a
nonzero
element
such
that
the
quotient
module
V
/R
·
v
is
free.
Then
the
centralizer
of
v
in
U
R
(V
)
is
equal
to
the
R-submodule
of
U
R
(V
)
generated
by
the
nonnegative
powers
of
v.
In
particular,
if
R
is
a
field
of
characteristic
zero,
then
the
centralizer
of
v
in
Lie
R
(V
)
is
equal
to
R
·
v.
(iii)
Suppose
that
the
rank
of
V
over
R
is
≥
2.
Then
the
Lie
algebra
Lie
R
(V
)
is
center-free.
In
particular,
the
adjoint
representation
of
Lie
R
(V
)
is
faithful.
(iv)
Let
R
be
an
R-algebra
which
is
finitely
generated
and
free
as
an
R-
module.
Let
φ
:
R
R
be
a
surjection
of
R-algebras;
suppose
that
V
=
V
⊗
R
,φ
R,
for
some
finitely
generated
free
R
-module
V
[so
we
obtain
a
natural
surjection
V
V
compatible
with
φ].
Then
the
natural
surjection
V
V
induces
a
sur-
jection
of
R-modules
Lie
bR
(V
)
Lie
bR
(V
)
that
factors
as
a
composite
of
natural
surjections
as
follows:
Lie
bR
(V
)
Lie
bR
(V
)
Lie
bR
(V
)
Here,
the
first
arrow
of
this
factorization
is
the
arrow
naturally
induced
by
observ-
ing
that
every
Lie
algebra
over
R
naturally
determines
a
Lie
algebra
over
R;
the
second
arrow
of
this
factorization
is
the
arrow
functorially
induced
by
the
natural
φ-
compatible
surjection
V
V
.
Finally,
this
second
arrow
induces
an
isomorphism
∼
Lie
bR
(V
)
⊗
R
,φ
R
→
Lie
bR
(V
).
Proof.
Assertion
(i)
follows
immediately
from
[Bour],
Chapter
II,
§2.11,
Theorem
1,
Corollary.
Assertion
(ii)
follows
from
the
well-known
structure
of
the
enveloping
algebra
U
R
(V
)
[i.e.,
the
natural
isomorphism
of
U
R
(V
)
with
the
free
associative
algebra
determined
by
V
over
R;
the
fact
that
when
R
is
a
field
of
characteristic
zero,
the
image
of
Lie
R
(V
)
in
U
R
(V
)
may
be
identified
with
the
set
of
primitive
ele-
ments
—
cf.
[Bour],
Chapter
II,
§3,
Theorem
1,
Corollaries
1,2],
by
considering
the
effect
on
“words”
of
forming
the
commutator
with
v
—
cf.
the
argument
of
[Mtm],
Proposition
3.1
[which
is
given
only
in
the
case
where
R
is
a
field
of
characteristic
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
81
zero,
but
does
not,
in
fact,
make
use
of
this
assumption
on
R
in
an
essential
way].
Assertion
(iii)
follows
immediately
from
assertion
(ii)
[by
allowing
the
element
“v”
of
assertion
(ii)
to
range
over
the
elements
of
an
R-basis
of
V
].
Assertion
(iv)
fol-
lows
formally
from
the
universal
property
of
a
free
Lie
algebra,
together
with
the
well-known
functoriality
of
a
free
Lie
algebra
with
respect
to
tensor
products
[cf.
[Bour],
Chapter
II,
§2.5,
Proposition
3].
Next,
let
us
suppose
that
we
have
been
given
data
as
follows:
(i)
a
prime
number
l;
(ii)
a
profinite
group
Δ
that
admits
an
normal
open
subgroup
V
⊆
Δ
such
that
the
following
conditions
are
satisfied:
(a)
V
is
abelian
[so
we
shall
regard
V
as
a
module];
(b)
the
topological
module
V
is
a
finitely
generated
def
free
R-module,
where
we
write
R
=
Z
l
;
(c)
the
resulting
action
of
the
def
def
finite
group
G
=
Δ/V
on
V
determines
a
G-module
V
Q
l
=
V
⊗
Q
l
that
contains
the
regular
representation
of
G;
(iii)
a
positive
power
N
of
l;
(iv)
a
collection
of
[not
necessarily
distinct!]
elements
g
1
,
.
.
.
,
g
d
∈
G
[where
d
≥
1
is
an
integer]
of
G
at
least
one
of
which
is
not
equal
to
the
identity
element.
Write
def
ζ
=
d
(1
−
g
i
)
∈
R[G]
i=1
[where
R[G]
is
the
group
ring
of
G
with
coefficients
in
R].
Then
we
have
the
following
result:
Lemma
A.2.
(Nontriviality
of
a
Certain
Operator)
There
exists
an
integer
n
≥
1
such
that
the
order
|J
ζ
|
of
the
image
J
ζ
⊆
J
def
of
the
action
of
ζ
on
[the
finite
group]
J
=
V
⊗
(Z/l
n
Z)
is
divisible
by
N
.
Proof.
Indeed,
since
the
G-module
V
Q
l
contains
the
regular
representation
[cf.
condition
(ii),
(c)],
it
follows
that
the
image
of
the
action
of
ζ
on
V
Q
l
is
a
nonzero
Q
l
-vector
space,
hence
that
the
image
of
the
action
of
ζ
on
the
finitely
generated
free
R-module
V
[cf.
condition
(ii),
(b)]
contains
a
rank
one
free
R-module.
Now
Lemma
A.2
follows
immediately.
def
Next,
let
J
ζ
⊆
J
be
as
in
Lemma
A.2;
write
H
=
Δ/(l
n
·
V
)
[so
J
⊆
H,
H/J
=
G].
Also,
let
us
assume
that
we
have
been
given
data
as
follows:
82
SHINICHI
MOCHIZUKI
(v)
a
collection
of
elements
h
1
,
.
.
.
,
h
d
∈
H
that
lift
g
1
,
.
.
.
,
g
d
∈
G;
(vi)
a
G-torsor
E
G
[whose
G-action
will
be
written
as
an
action
from
the
left];
(vii)
an
H-torsor
E
H
[whose
H-action
will
be
written
as
an
action
from
the
left];
(viii)
a
surjection
:
E
H
E
G
that
is
compatible
with
the
natural
surjection
H
G;
(ix)
a
continuous
action
of
a
profinite
group
Γ
on
Δ
that
preserves
the
sub-
def
group
V
⊆
Δ,
hence
determines
a
profinite
group
Δ
Γ
=
Δ
Γ
that
acts
continuously
on
G,
H
[in
such
a
way
that
the
restriction
of
this
action
to
Δ
⊆
Δ
Γ
is
the
action
of
Δ
on
G,
H
by
conjugation];
(x)
continuous
actions
of
Δ
Γ
on
E
G
,
E
H
[which
will
be
denoted
via
super-
scripts]
that
are
compatible
with
the
continuous
actions
of
Δ
Γ
on
G,
H,
as
well
as
with
the
surjection
and,
moreover,
induce
the
trivial
action
of
Γ
⊆
Δ
Γ
on
E
G
[hence
also
on
G];
(xi)
an
element
[i.e.,
“basepoint”]
e
H
∈
E
H
,
whose
image
via
we
denote
by
e
G
∈
E
G
,
such
that
e
H
,
e
G
are
fixed
by
the
action
of
Δ
Γ
on
E
H
,
E
G
.
Next,
let
us
write
def
R
J
=
R[J
]
for
the
group
ring
of
J
with
coefficients
in
R.
Thus,
R
J
is
a
commutative
R-algebra,
and
we
have
a
natural
augmentation
homomorphism
R
J
R
[which
sends
all
of
the
elements
of
J
to
1].
Moreover,
if
we
write
def
def
M
:
M
H
=
R[E
H
]
M
G
=
R[E
G
]
for
the
morphism
of
R
J
-modules
induced
by
on
the
respective
free
R-modules
with
bases
given
by
the
elements
of
E
H
,
E
G
,
then
M
induces
a
natural
isomorphism
∼
M
H
⊗
R
J
R
→
M
G
.
Thus,
it
follows
from
Proposition
A.1,
(iv),
that,
for
b
≥
1
an
integer,
we
have
[in
the
notation
of
Proposition
A.1]
natural
surjections
Lie
bR
(M
H
)
Lie
bR
J
(M
H
)
Lie
bR
(M
G
)
∼
the
second
of
which
determines
a
natural
isomorphism
Lie
bR
J
(M
H
)⊗
R
J
R
→
Lie
bR
(M
G
).
Now
let
P
(X
1
,
.
.
.
,
X
d
)
be
an
“alternant
monomial
of
degree
d”
[i.e.,
a
monomial
element
of
Lie
d
Z
(−)
of
the
free
Z-module
on
the
indeterminate
symbols
X
1
,
.
.
.
,
X
d
]
in
which
each
X
i
[for
i
=
1,
.
.
.
,
d]
appears
precisely
once.
Then
P
(X
1
,
.
.
.
,
X
d
)
determines
an
element
P
(g
1
·
e
G
,
.
.
.
,
g
i
·
e
G
,
.
.
.
,
g
d
·
e
G
)
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
83
of
Lie
dR
(M
G
).
Moreover,
by
allowing
such
P
(X
1
,
.
.
.
,
X
d
)
and
g
1
,
.
.
.
,
g
d
to
vary
ap-
propriately,
we
obtain
a
Hall
basis
[cf.,
e.g.,
[Bour],
Chapter
II,
§2.11]
of
Lie
dR
(M
G
)
[at
least
if
d
≥
2;
if
d
=
1,
then
one
must
also
allow
for
the
unique
g
1
to
be
the
identity
element].
Similarly,
by
allowing
such
P
(X
1
,
.
.
.
,
X
d
)
and
h
1
,
.
.
.
,
h
d
∈
H
to
vary
appropriately,
we
obtain
a
Hall
basis
[again,
strictly
speaking,
if
d
≥
2]
of
Lie
dR
J
(M
H
)
of
elements
of
the
form
P
(h
1
·
e
H
,
.
.
.
,
h
d
·
e
H
).
Lemma
A.3.
(Relation
of
Superscript
and
Left
Actions)
For
any
v
∈
V
⊆
Δ
⊆
Δ
Γ
that
maps
to
j
∈
J,
we
have
P
(h
1
·
e
H
,
.
.
.
,
h
i
·
e
H
,
.
.
.
,
h
d
·
e
H
)
v
=
ζ(j)
·
P
(h
1
·
e
H
,
.
.
.
,
h
i
·
e
H
,
.
.
.
,
h
d
·
e
H
)
in
Lie
dR
J
(M
H
).
Proof.
Indeed,
we
compute:
P
(h
1
·
e
H
,
.
.
.
,h
i
·
e
H
,
.
.
.
,
h
d
·
e
H
)
v
=
P
(h
v
1
·
e
H
,
.
.
.
,
h
vi
·
e
H
,
.
.
.
,
h
vd
·
e
H
)
−1
−1
v
v
=
P
(h
v
1
·
h
−1
1
·
h
1
·
e
H
,
.
.
.
,
h
i
·
h
i
·
h
i
·
e
H
,
.
.
.
,
h
d
·
h
d
·
h
d
·
e
H
)
=
d
[j,
h
i
]
·
P
(h
1
·
e
H
,
.
.
.
,
h
i
·
e
H
,
.
.
.
,
h
d
·
e
H
)
i=1
=
ζ(j)
·
P
(h
1
·
e
H
,
.
.
.
,
h
i
·
e
H
,
.
.
.
,
h
d
·
e
H
)
[where
we
apply
the
R
J
-module
structure
of
E
H
and
the
fact
that
e
vH
=
e
H
[cf.
(xi)]].
Next,
let
us
assume
that
we
have
also
been
given
the
following
data:
(xii)
a
topological
R-module
Λ
equipped
with
a
continuous
action
by
Γ
[which
thus
determines,
via
the
natural
surjection
Δ
Γ
Γ,
a
continuous
action
by
Δ
Γ
on
Λ].
Write:
def
V
Γ
=
V
Γ
⊆
Δ
Γ
;
def
F
J
=
J
·
P
(h
1
·
e
H
,
.
.
.
,
h
d
·
e
H
)
⊆
Lie
dR
J
(M
H
);
def
R[F
J
]
=
R
·
F
J
=
R
J
·
P
(h
1
·
e
H
,
.
.
.
,
h
d
·
e
H
)
⊆
Lie
dR
J
(M
H
);
def
Λ[F
J
]
=
R[F
J
]
⊗
R
Λ
⊆
Lie
dR
J
(M
H
)⊗
R
;
Λ
def
F
=
P
(g
1
·
e
G
,
.
.
.
,
g
d
·
e
G
)
∈
Lie
dR
(M
G
);
def
R[F
]
=
R
·
F
⊆
Lie
dR
(M
G
);
def
Λ[F
]
=
R[F
]
⊗
R
Λ
⊆
Lie
dR
(M
G
)
⊗
R
Λ
Thus,
the
natural
surjection
Lie
dR
J
(M
H
)
Lie
dR
(M
G
)
determines
[compatible]
nat-
ural
surjections
F
J
{F
},
R[F
J
]
R[F
],
Λ[F
J
]
Λ[F
].
Also,
we
observe
[cf.
the
84
SHINICHI
MOCHIZUKI
fact
that
Lie
dR
J
(M
H
)
is
a
finitely
generated
free
R
J
-module]
that
F
J
is
a
J-torsor
[relative
to
the
action
from
the
left],
hence,
in
particular,
a
finite
set.
Now
observe
that
since
V
Γ
acts
trivially
on
G,
e
H
[cf.
(ix),
(x),
(xi)],
it
follows
immediately
that
V
Γ
acts
compatibly
on
F
J
,
R[F
J
],
Λ[F
J
],
F
,
R[F
],
Λ[F
],
and
that
the
natural
action
of
V
Γ
on
R[G]
preserves
ζ.
In
particular,
it
follows
that
V
Γ
preserves
J
ζ
⊆
J,
hence
that
V
Γ
acts
naturally
on
the
set
of
orbits
(F
J
)
F
ζ
of
F
J
with
respect
to
the
action
of
J
ζ
;
moreover,
by
Lemma
A.3,
it
follows
that
this
action
of
V
Γ
on
F
ζ
factors
through
the
quotient
V
Γ
Γ.
Now
let
us
consider
invariants
with
respect
to
the
various
superscript
actions
under
consideration.
Let
us
write
Invar(−,
−)
for
the
set
of
invariants
of
the
second
argument
in
parentheses
with
respect
to
the
superscript
action
of
the
group
given
by
the
first
argument
in
parentheses.
Then
any
element
η
∈
Invar(V
Γ
,
Λ[F
J
])
may
be
regarded
as
a
Λ-valued
function
on
the
set
F
J
which
descends
[cf.
Lemma
A.3]
to
a
Γ-invariant
Λ-valued
function
on
F
ζ
,
i.e.,
an
element
η
ζ
∈
Invar(Γ,
Λ[F
ζ
]).
Next,
let
us
observe
that
[since
η
ζ
is
Γ-invariant]
the
sum
of
the
values
∈
Λ
of
the
Λ-
valued
function
on
F
ζ
determined
by
η
ζ
is
a
Γ-invariant
element
η
ζ
∈
Invar(Γ,
Λ).
Thus,
the
sum
η
∈
Λ
of
the
values
∈
Λ
of
the
Λ-valued
function
on
F
J
determined
by
η
satisfies
the
relation
η
ζ
η
=
|J
ζ
|
·
in
Λ.
But
the
image
of
η
in
Λ[F
]
is
precisely
the
element
(
η)
·
F
.
Thus,
since,
by
Lemma
A.2,
|J
ζ
|
is
divisible
by
N
,
we
conclude
the
following:
Lemma
A.4.
(Monomial-wise
Computation
of
Invariants)
The
image
Im(Invar(V
Γ
,
Λ[F
J
]))
⊆
Λ[F
]
of
Invar(V
Γ
,
Λ[F
J
])
⊆
Λ[F
J
]
in
Λ[F
]
lies
in
N
·
Invar(Γ,
Λ[F
]).
Thus,
by
allowing
P
(X
1
,
.
.
.
,
X
d
)
and
h
1
,
.
.
.
,
h
d
∈
H
as
in
the
above
discus-
sion
to
vary
appropriately
so
as
to
obtain
a
Hall
basis
[again,
strictly
speaking,
if
d
≥
2]
of
Lie
dR
J
(M
H
)
of
elements
of
the
form
P
(h
1
·
e
H
,
.
.
.
,
h
d
·
e
H
),
we
conclude
the
following:
ABSOLUTE
ANABELIAN
CUSPIDALIZATIONS
85
Theorem
A.5.
(Invariants
of
Free
Lie
Algebras)
Let
d
≥
1
be
an
integer.
Suppose
that
we
have
been
given
data
as
in
(i),
(ii),
(iii)
above.
Let
n
≥
1
be
an
integer
that
satisfies
the
property
of
Lemma
A.2
for
all
[of
the
finitely
many]
def
possible
choices
of
data
as
in
(iv)
[relative
to
the
given
integer
d
≥
1];
J
=
V
/(l
n
·
def
def
V
)
⊆
H
=
Δ/(l
n
·
V
);
R
J
=
R[J
].
Suppose
that
have
also
been
given
data
as
def
def
in
(vi),
(vii),
(viii),
(ix),
(x),
(xi),
(xii)
above;
let
M
H
=
R[E
H
],
M
G
=
R[E
G
],
def
V
Γ
=
V
Γ
(⊆
Δ
Γ
).
Then
the
natural
surjection
Lie
dR
J
(M
H
)
⊗
R
Λ
Lie
dR
(M
G
)
⊗
R
Λ
maps
Invar(V
Γ
,
Lie
dR
J
(M
H
)
⊗
R
Λ)
into
N
·
Invar(V
Γ
,
Lie
dR
(M
G
)
⊗
R
Λ)
if
d
≥
2.
In
a
similar
vein,
the
natural
surjection
M
H
⊗
R
Λ
M
G
⊗
R
Λ
maps
Invar(V
Γ
,
M
H
⊗
R
Λ)
into
N
·
Invar(V
Γ
,
M
G
⊗
R
Λ)
+
Invar(V
Γ
,
Λ)
·
e
G
⊆
M
G
⊗
R
Λ.
86
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